la

Simof Taoad

Available online 16 February 2012

Keywords:Composite laminated Reddy plateModied couple stressMaterial length parameterScale effect

ple

ories based on the modied couple stress theory by Yang et al. in 2002. The form of new curvature tensoris asymmetric, however it can result in same as the symmetric curvature tensor in the isotropic elasticity.

increases by a factor of three as the wire diameter decreases from170 lm to 12 lm in the twisting of thin copper wires; Lam et al. [2]reported a signicant increase in the normalized bending harden-ing with the beam thickness decreasing in bending of ultra thinbeams. Sun et al. [3] put forward a alternative view of the sizeeffects in the nano-scale structures. As conventional continuum

For C theory, the displacements and rotations/strains are indepen-dent variables. The C0 couple stress theory (Cosserat-type theories )which contains more than two additional material constants. Ingeneral, the Cosserat-type theories have six additional micro-material constants. The C0 couple stress theory have to have twoadditional micro-material constants, however, the C1 couple stresstheory may have one additional micro-material constants. TheC1 couple stress theory cannot explain as pseudo-Cosseratapproach. It is well known that in the classical bending platetheory, the thin plate theory is called as C1 theory, themediumplatetheory is called as C0 theory. Both classical bending plate theories

Corresponding author at: Key Laboratory of Liaoning Province for CompositeStructural Analysis of Aerocraft and Simulation, Shenyang Aerospace University,Shenyang, LN 110136, China.

Composite Structures 94 (2012) 21432156

Contents lists available at

Composite S

sevE-mail address: chenwj@dlut.edu.cn (W.J. Chen).1.1. Scale effects of microstructure

With the material size scaling down to the order of micro-scales, the stiffness and the strength of metal materials canincrease with the size decreasing, which is called size effects. Themicrostructure-dependent size effects have been exhibited bymanymicro- and nano-scale components and devices. The classicalelasticity theory is not capable of predicting such size effects. Toovercome this deciency, theories for microstructures need to bedeveloped.

Numerous experiments have shown that microstructure hasscale effects. Fleck et al. [1] observed that the scaled shear strength

1.2. Couple stress theory

Theories for microstructures include couple stress theory andstrain gradient theory. The couple stress theory can be viewed asa special case of strain gradient theory which uses rotation as a var-iable to describe curvature, while the strain gradient theory usesstrain as a variable to describe curvature. Though both theoriescan describe the scale defects at micro-scale The couple stress/strain gradient theory for microstructures can be classied intotwo respective theories, C1 theory and C0 theory. For C1 theorythe displacements and rotations/strains are dependent variables.

01. Introduction0263-8223/$ - see front matter 2012 Elsevier Ltd. Adoi:10.1016/j.compstruct.2012.02.009The present model of thick plate can be viewed as a simplied couple stress theory in engineeringmechanics. Moreover, a more simplied model for cross-ply composite laminated Reddy plate of couplestress theory with one materials length constant is used to demonstrate the scale effects. Numericalresults show that the present plate model can capture the scale effects of microstructure. Additionally,the present model of thick plate model can be degenerated to the model of composite cross-ply laminatedKirchhoff plate and Mindlin plate of couple stress theory.

2012 Elsevier Ltd. All rights reserved.

theory cannot explain or solve the problems of the scale effects,theories for microstructures need to be developed.rst time. In this theory a new curvature tensor is dened for establishing the constitutive relations oflaminated plate. The characterization of anisotropy is incorporated into higher-order laminated plate the-A model of composite laminated Reddy pstress theory

Wanji Chen a,b,, Ma Xu b,c, Li Li b,daKey Laboratory of Liaoning Province for Composite Structural Analysis of Aerocraft andb State Key Laboratory for Structural Analysis of Industrial Equipment, Dalian Universityc School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, Chind Physics 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:

a b s t r a c t

Based on newmodied cou

journal homepage: www.elll rights reserved.te based on new modied couple

ulation, Shenyang Aerospace University, Shenyang, LN 110136, Chinaechnology, Dalian 116023, China

, Heping District, Shenyang 110001, China

stress theory a model for composite laminated Reddy plate is developed in

SciVerse ScienceDirect

tructures

ier .com/locate /compstruct

truchave a distinction to have connection again, when the thinness ofplate became very thin, the C0 theory can approximate C1 theory.Unlike classical bending plate theory, the C1 theory and C0 theoryfor couple stress theory cannot nd a connection each order.

For the couple stress theory, the C1 theory have to satisfy therelation of rotation-displacement as xi 12 uk;j uj;k 0: How-ever, the C0 theory cannot satisfy this condition and this termhas to possess one independent micro-material constant. Up todayit is no answer about the condition for vanishing this term in the C0

the couple stress theory.A series of research in the couple stress theories have been

made. For example, Toupin [4], Koiter [5] and Mindlin [6] proposedcouple stress theory; Neuber [7] proposed couple stress theorycontaining four materials characteristic length constants; FleckHutchinson [8] proposed couple stress theory containing onematerials characteristic length constants; Yang [9] proposed sym-metrical couple stress theory (C1 theory) containing one materialcharacteristic length constants.

The microstructures such as sensors and actuators in micro-electromechanical systems (MEMS) and nano-electromechanicalsystems (NEMS) often consist in the components of beam, plateand membrane et al. According to the application in engineering,the beam, plate and shell theories based on couple stress/straingradient theory should be developed.

1.3. The couple stress isotropic plate model based on the C0 couplestress theory

The researchers have focused on the plate theory on micro-scalein recent years. A number of papers have been published forattempting to develop microstructure-dependent non-local Timo-shenko plate models and apply them to analyze nanotubes andother small plate-like members/devices. All of these models arebased on a C0 theory in which the rotation-displacement as depen-dent variables. For example, the model for pure bending proposedby Anthoine [10] is based on the classical C0 couple stress elasticitytheory, which includes two additional internal material lengthscale parameters. The higher-order BernoulliEuler plate modeldeveloped by Papargyri-Beskou et al. [11] is based on the C0 gradi-ent elasticity theory, which involves two internal material lengthscale parameters. The non-local BernoulliEuler plate model pro-posed by Peddieson et al. [12], in the formulation the constitutiveequation suggested by Eringen [13] contains two additional mate-rial constants. More background related to the couple stress platebased on the C0 couple stress theory, especially Cosserat-type the-ories which contain more than two additional material constants,can be found in the review by Johannes al. [14].

1.4. The couple stress isotropic plate model based on the C1 couplestress theory

Recently, due to the difculty of determining more than onemicrostructure-dependent length scale parameters and theapproximate nature of plate theories, C1 non-classical plate modelsinvolving 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 [15]. Ma et al. [16] proposed a microstructure-dependent Tim-oshenko beam model based on a modied couple stress theorywith only one material length parameter. Tsiatas [17] proposed anew Kirchhoff plate model based on a modied couple stresstheory, and by using modied couple stress theory, the governing

2144 W.J. Chen et al. / Composite Sequation of couple stress thin plate are obtained as(D + 2D)r4w = fw, where D and Dl are the bending rigidity androtation gradients rigidity of the plate respectively [17]. This equa-tion can show the scale effects obviously, so many researchers getmany attentions to use the modied couple stress theory to studythe couple stress plate and beam models. Metin [18] developed ageneral nonlocal beam theory based on C0 theory where the non-local constitutive equations proposed by Eringen [13] are adopted.The nonclassical ReddyLevinson (RL) beam model based on thehigher order shear deformation theory and C1 couple stress theorywas developed by Ma et al. [19]. The non-classical RL model canbe reduced to the existing classical elasticity-based RL model byusing the material length scale parameter and Poissons ratio areboth taken to be zero. The classical RL beam model [20] is athird-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 plate model can be reduce thenon-classical BernoulliEuler beam model when the normalityassumption is introduced. Recently, Li et al. used directly the mod-el of modied couple stress theory proposed by Tsiatas [16] toanalysis of vibration of micro-scale plates [21]. Jomehzadeheiet al. based on a modied couple stress theory to analysis of thesize-dependent vibration of micro-plates [22]. Ma et al. based ona modied couple stress theory establish a Mindlin plate model[23]. Wang et al. based on strain gradient elasticity theory pro-posed a Kirchhoff micro-plate model [24]. Recently, Reddy et al.developed models of functionally graded beams and nonlinear for-mulations based on nonlocal/couple stress theory. The couplestress theory is used to analysis functionally graded beams by Red-dy [25]. The nonlocal nonlinear formulations for beams and plateswas developed by Reddy [26] and Reddy et al. based on modiedcouple stress proposed a nonlinear third-order theory of function-ally graded plates [27].

Today, by using modied couple stress theory to establish themodel of beam and plate in the microstructures is just starting.The rst paper is published in 2006, however, up to 2011 relativemany papers have been published. A hot point to study the mi-cro-shale effects may arise.

Existing modied couple stress theory belong to isotropic the-ory, so the plate models established based on the modied couplestress theory belong to isotropic theory including the non-classicalMindlin plate model based on a modied couple stress theory [23].These models and existing modied couple stress theory cannotapply to establish anisotropic plate model, especially the study ofcouple stress laminated plate theory.

Composite laminate plates are widely used in engineering. Dueto the micro-scale impurities such as bre, and microcracks atmicro-matrix are involved in a laminated composite structures, itresults in classical laminate theory invalid in some problemsrelated to the micro-scale of laminate composites.

It is well known that the constitute relation of isotropic elastic-ity can be easily extended to anisotropic elasticity. Unlike the clas-sical elasticity theory, for the modied couple stress theory, thisextension cannot be easy, especially composites laminated plate.In the couple stress theory, the rotation variables related to mi-cro-scale impurities or defects are formulated into rotation equilib-rium equations. The anisotropic elasticity of the couple stresstheory depends on the single component of rotation rather thanthe assembly of derivative of rotations. The symmetric curvaturedened on the modied couple stress theory is an assembly ofthe derivative of rotations, so it cannot be easily extended toanisotropic.

The study for couple stress laminated plate is presented in rsttime in this paper. Firstly the isotropic modied couple stress the-ory is extended to anisotropic modied couple stress theory and it

tures 94 (2012) 21432156can degenerate to the isotropic modied couple stress theory. Thecontribution of this work is that a new curvature tensor is denedfor establishing the constitutive relations of laminated plate as

anisotropy materials. Secondly, the model of composite laminatedReddy plate of couple stress theory and analysis of the scale ef-cient are given in the rst time.

2. Formulations for composite laminated plate based on newmodied couple stress theory

The main differences of modied couple stress theory with thestandard couple stress theory are that for modied couple stresstheory the couple stress tensor is symmetric and only one internalmaterial length scale parameter is considered [9], however, forstandard couple stress theory, the couple stress tensor is asymmet-ric and number of internal material length scale parameters is onenot always.

2.3. Hystress

Inby int

placemcoupleFig. 1,basedxz is amodelwe ashigh-otheory.

(1)

W. Chen et al. / Composite Structures 94 (2012) 21432156 21452.1. Classical couple stress theory

Mindlin developed a couple stress theory [6], which can becalled as classical couple stress theory.

The strain tensor and curvature tensor can be dened respec-tively as

eij 12 ui;j uj;ivij xi;j

(2-1

where ui the displacement vector, xi is the rotation vector andxi 12 eijkuk;j; eij is symmetric tensor and vij is asymmetric tensor.

Constitutive relations are given by

rij kekkdij 2Geijmij 42Gvij

(2-2

where k and G are elasticity constants, is the materials constant ofthe microstructures.

2.2. Modied couple stress theory

Unlike the standard couple stress theory, Yang et al. [9] devel-oped modied couple stress theory in which the part of rotationgradient in the strain tensor is symmetric.

According to the symmetric couple stress theory, the strain ten-sor and curvature tensor can be dened respectively as

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

(2-3

where x 12curl u;uui is the displacement vector and x(xi) isthe rotation vector, eij and vij are symmetric tensor.

The strain energy can be written as

U 12

ZXr : em : vdv 2-4

where r stress tensor, strain e tensor,m couple stress moment ten-sor and v curvature tensor. They are dened by

r ktreI 2Gem 22Gv

2-5

In (2-2) the coefcient 4 is chosen for a couple stress theorydeveloped by Mindlin [6]. However, in (2-5) the coefcient 2 ischosen for a modied couple stress theory developed by Yanget al. [9], which can ensure the coefcients of the constitutiverelations the couple stress tensor is symmetric. It is means thatthe value of in the two theories is difference only with a multiple.Fig. 1. Schematic diagram of cvx; y; z v0x; y z @w@y

w wx; y

2-6

(2) The medium thickness plate: the Mindlin plate theory. Thedisplacement eld is assumed as

ux; y; z u0x; y zhyx; yvx; y; z v0x; y zhxx; yw wx; y

2-7

where according to the engineering conventional representa-tion, hx, hy are the angles of rotations around the y, x axes ofthe cross-section, and for Kirchhoff plate: hx @w@y ; hy @w@x .

(3) The thick plate: the Reddy plate theory. The Reddy plate the-ory, known as a third-order plate theory, is based on the dis-placement eld

ux; y; z u0x; y zhyx; y cz3 hy @wdx

vx; y; z v0x; y zhxx; y cz3 hx @wdy

wx; y; z w0x; y

2-8ompositThe thin plate: the Kirchhoff plate theory. The displacementeld is assumed as

ux; y; z u0x; y z @w@x2.4. Two-dimensional plate theoriesents along x, y and z directions respectively. For the platestress theory, xx, xy and xz are the rotations as shown inwhile it is assumed as xz = 0 for the Kirchhoff plate modelon modied couple stress theory [17], however, the rotationdopted in the formulation of the non-classical Mindlin platebased on a modied couple stress theory [23]. In this papersume xz = 0 for modeling composite laminated plate withrder shear deformation based on the modied couple stressis also true for the composite laminated plate for the couple stresstheory.

The displacements are represented by u, v and w, which are dis-pothesis of composite laminated plate of new modied coupletheory

the point of elastic theory, the plate theory can be describedroducing the hypothesis of the cross-section into the plate. Ite laminated plate.

truc2.5. Strain for modied couple stress theory of isotopic elasticity

The strain tensor and curvature tensor for modied couplestress theory of isotopic elasticity can be written as

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

(2-10

According to the engineering conventional representation, thestrain and curvature for the plate in isotropic elasticity can be ex-pressed as follows:

ex u;xey v ;y; cxy 2c12; vx v11; vy v22;vxy 2v12:Due to xz = 0, we have

vxzvyz

@xz@x @xx@z

@xz@y @xy@z

( ) 0

The strain can be written as

e

exeycxycxzcyz

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

9>>>>>>>=>>>>>>>;

@u@x

@v@y

@u@y @v@x@w@x @u@z@w@y @v@z

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

9>>>>>>>>>=>>>>>>>>>;2-11-1

and

v vxvyvxy

8>:9>=>;

@xx@x

@xy@y

@xx@y @xy@x

8>>>:9>>=>>; 2-11-2

2.6. Constitutive relations for composite laminated plate based on newmodied couple stress theory

The constitutive relations of the anisotropic elasticity of thecouple stress theory depend on the component of rotation. Forthe composite laminated plate, the @xx

@y and@xy@x should be taken as

independent components which are relative to the micro-materials constants 2b ;

2m to describe the ber and matrix of the

laminated plate respectively.The new expression of the constitutive relations can be written

as follows

mx

my

8>>>>9>>>>=>

2C442b

2C552m2 2

266666377777

@xx@x

@xy@y

@xx

8>>>>>>9>>>>>>=> 2-12where c 4l3h2

;l is a parameter to control model: l = 0rst ordermodel, l = 1 Reddy model, hx, hy are the angle of rotation aroundthe y-, x-axis of the cross-section respectively (see Fig. 1). The dis-placement elds shown in (2-6)(2-8) can be found in Book [28].

Substituting Eq. (2-8) into the expression of the rotation asx 12 curl u, we have,

xx 12 w;y v ;z 121 3cz2w;y 1 3cz2hx

xy 12 u;z w;x 121 3cz2w;x 1 3cz2hy

xz 12 v ;x u;y 0

2-9

where c 4l3h2

and a comma followed by a subscript denotes differ-entiation with respect to the subscript (e.g., u,x = @u/@x).

2146 W.J. Chen et al. / Composite Smxy

myx

>>>: >>>; C44b C55mC442b C55

2m

4 5 @y@xy@x

>>>>>: >>>>>;Substituting Eqs. (2-8) and (2-9) into (2-11-2), (2-12), the strainof the laminated plate can be obtained as follows:

em

exeycxycxzcyz

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

9>>>>>>=>>>>>>;

u0;x cz3w;xxz cz3hy;xv0;y cz3w;yy z cz3hx;yu0;y v0;x 2cz3w;xy z cz3hy;y z cz3hx;x13cz2w;x hy13cz2w;y hx

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

9>>>>>>=>>>>>>;2-13

and

v

vxvyvxyvyx

8>>>>>:9>>>=>>>;

12 1 3cz2w;xy 1 3cz2hx;x 12 1 3cz2w;xy 1 3cz2hy;y12 1 3cz2w;yy 1 3cz2hx;y 12 1 3cz2w;xx 1 3cz2hy;x

8>>>>>:9>>>=>>>; 2-14

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

The stressstrain relations of kth layer in the local coordinate(x0,y0,z0) can be expressed as follows:

rk Ckek 2-15where

rk rkx0 rky0 skx0y0 skx0z0 sky0z0 mkx0 mky0 mkx0y0 mky0x0h iT

ek ekx0 eky0 ckx0y0 ckx0z0 cky0z0 vkx0vky0 vkx0y0 vky0x0h iT

8>:2-16

Ck

Ck11 Ck12

Ck21 Ck22

Ck66Ck44

Ck5522bC

k44

22mCk55

2bCk44

2mC

k55

2bCk44

2mC

k55

266666666666666666664

3777777777777777777752-17

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

Ck11 1vk22vk22Ek2E

k2D

, Ck12 Ck21 vk21vk22vk21

Ek2Ek2D

, Ck22 1vk21vk12Ek1E

k2D

; Ck44 Gk12;Ck55

Gk22, Ck66 Gk12, vk21

Ek2vk12

Ek1, D 1v

k12v

k21vk22vk22vk12vk212vk21vk22vk12

Ek1Ek2E

k2

and

Ek1; Ek2

; Gk12;G

k22

and vk12;vk21

are the elastic constants, shear

elastic constants and Poisson ratios of kth layer respectively, inwhich subscripts 1 and 2 represent the direction of ber and matrix.2b ;

2m are constants to describe the materials microstructural char-

acteristics of respectively.After coordinate transformation, the stressstrain relations of

kth layer in the global coordinate (x,y,z) can be written as follows

rk Q ke 2-18where

rk rkx rky skxy skxz skyz mkx mkymkxymkyxh iT

e ex ey cxy cxz cyzvxvyvxyvyx

T

8

< =@xy

< =

tructThe coordinate transformation matrix, Tk, is expressed as

Tk Tk1

Tk2

" #2-21

Tk1 m2 n2 mn

n2 m2 mnmn mn m2 n2

264375; Tk2 T0 T1

;

T0 m n

n m

; T1

m2 n2 mn mn

n2 m2 mn mnmn mn m2 n2mn mn n2 m2

2666437775 2-22

where /k is angle of ply and m = cos/k, n = sin/k.The components of Qk are expressed as

Q k Qkm

Q kc

" #2-23

Q k

Qk11 Qk12 Q

k16

Qk12 Qk22 Q

k26

Qk16 Qk26 Q

k66

Qk44 Qk45

Qk45 Qk55

22 eQk44 22 eQk45 22 eQk46 22 eQ k4722 eQk45 22 eQk55 22 eQk56 22 eQ k5722 eQk57 22 eQk46 2 bQ k44 2 bQk5522 eQk57 22 eQk46 2 bQ k44 2 bQk55

266666666666666666664

3777777777777777777752-24

where

Qk11 m4Ck11 n4Ck22 2m2n2Ck12 4m2n2Ck66Qk22 n4Ck11 m4Ck22 2m2n2Ck12 4m2n2Ck66Qk12 m2n2 Ck11 Ck22 4Ck66

m4 n4Ck12

Qk16 m3n Ck11 Ck22 2Ck66

mn3 Ck12 Ck22 2Ck66

Qk26 mn3 Ck11 Ck22 2Ck66

m3n Ck12 Ck22 2Ck66

Qk66 m2n2 Ck11 Ck22 2Ck12

m2 n22Ck66Qk44 Ck44m2 Ck55n2Qk45 mn Ck44 Ck55

Qk55 Ck44n2 Ck55m2

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:2-25

2 eQk44 2bCk44m4 2mCk55n4 4m2n2 2bCk44 2mCk55 2 eQk45 3m2n2 2bCk44 2mCk55 2 eQk55 n42bCk44 m42mCk55 4m2n2 2bCk44 2mCk55 2 eQk46 mn m22bCk44 n22mCk55 2 eQk47 mnm2 2n22bCk44 mnn2 2m22mCk552 eQk56 mnn2 2m22bCk44 mnm2 2n22mCk552 eQk57 mn n22bCk44 m22mCk55 2 bQk44 2bCk44m4 2mCk55n4 m2n2 2bCk44 2mCk55

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

2-26

W. Chen et al. / Composite S2 bQk55 n42bCk44 m42mCk55 m2n2 2bCk44 2mCk55 >>>>:v vyvxy

>: >; @y@xx@y @xy@x

>>: >>; 2-27Constitutive relations to relative to the curvature are given by

m mxmymxy

8>:9>=>; 2G2

@xx@x

@xy@y

@xx@y @xy@x

8>>>:9>>=>>; 2-28

The strain energy of the part of curvature can be written as

U 12

ZXmv : v dv 2-29

For isotropic plate:

m : v 2G2@xx@x

@xy@y

@xx@y @xy@x

8>>>>>:

9>>>=>>>;T @xx

@x@xy@y

@xx@y @xy@x

8>>>>>:

9>>>=>>>; 2-30For new modied couple stress theory, constitutive relations to

relative to the curvature are given by

mxmymxymyx

8>>>>>:9>>>=>>>;

2C442b@xx@x

2C552m@xy@y

C442b@xx@y C552m @xy@x

C442b

@xx@y C552m @xy@x

8>>>>>>>>>:

9>>>>>=>>>>>;2-31

For isotropic plate: C44 = C55 = G and b = m = .

mxmymxymyx

8>>>>>:9>>>=>>>;

2G2 @xx@x

2G2 @xy@y

G2 @xx@y G2 @xy@x

G2 @xx@y G2 @xy@x

8>>>>>>>>>:

9>>>>>=>>>>>; 2G2

@xx@x

@xy@y

12

@xx@y @xy@x

12

@xx@y @xy@x

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

9>>>>>>=>>>>>>;2-32

The strain energy of the part of curvature can be written as

U 12

ZXmv : vdv 2-33

For isotropic plate:

m : v 2G2@xx@x

@xy@y

@xx@y @xy@x

8>>>>>:

9>>>=>>>;T @xx

@x@xy@y

@xx@y @xy@x

8>>>>>:

9>>>=>>>; 2-34Obviously, (2-30) is identical to (2-34), so formulations aniso-

tropic plate such as laminated plate and sandwich plate can beused to isotropic plate by using Qkij Ckij; b m .

3. Potential energy principle for composite laminated Reddyplate of new modied couple stress theory

It is well known that the potential energy principle can be usedto derive the equilibrium equation and the boundary condition.

The potential energy principle for composite laminated Reddywhere m = cos/k, n = sin/k.Aforementioned formulations can use to isotropic plate and

anisotropic plate such as laminated plate and sandwich plate.The modied couple stress theory proposed by Yang [9] is a the-

ory of isotopic elasticity. The curvature is written as follows:

vx8> 9> @xx@x8>> 9>>

ures 94 (2012) 21432156 2147plate of modied couple stress theory is given by

dpp dU dW 0 3-1

where pp is the total potential energy of the deformable body, U isthe strain energy of the deformable body, W is the work producedby the external forces, dU and dW are the rst variation of U andW respectively.

dU Xnk1

dUk Xnk1

ZVkrkTdedxdydz

ZX

Xnk1

Z zk1zk

rkTdedz

!dxdy

3 - 2dW

ZX

f Tdudv Z

@XTTduds 3-3

where f T and TT are the body force on the plate domain X andboundary force on the plate boundary oX, respectively.

The dW in the plate on the modied couple stress theory can beexpressed as

and

e0m e0xe0yc0xy

8>:9>=>;

u0;xv0;yu0;y v0;x

8>:9>=>;; e1m

e1xe1yc1xy

8>:9>=>;

hy;xhx;yhy;y hx;x

8>:9>=>;;

e3m e3xe3yc3xy

8>:9>=>; c

hy;x w;xxhx;y w;yyhy;y 2w;xy hx;x

8>:9>=>;

3-9-1

v0 c0xz c0yz v0x v0y v0xy v0yxh iT

12

2w;x hy;2w;y hx;

T

YyYx

2;xy

12

ndhn

;yy3

2148 W.J. Chen et al. / Composite Structures 94 (2012) 21432156dW ZX

fudu0 fvdv0 fwdw fcxdxx fcydxy

dxdy

Z

@XNnxdu0 Nnydv0 VdwMndhn

ds 3-4

where fu, fv and fw are, respectively, the x-, y- and z-components ofthe body force per unit length, fcx and fcy are the x- and y-componentof body couple force per unit length along the x- and y-axis, andNx;Ny;Nxy are the applied axial forces and shear force, V and Mntransverse force and bending moment at the boundary of the platerespectively, andR

X fcxdxxdxdy 12RX fcxdhx w;ydxdy 12

RXfcxdhx @fcx@y dwdxdy

12R@X fcxnydwdsR

X fcydxydxdy 12RX fcydhy w;xdxdy 12

RXfcydhy @fcy@x dwdxdy

12R@X fcynxdwds

8>>>>>>>:3-5

Substituting (3-5) into (3-4), we have

dW ZX

fudu0 fvdv0 fw 12@fcy@x

@fcx@y

dw

12fcxdhx fcydhy

dxdy

Z@XNnxdu0 Nnydv0

V 12fcxny fcynx

dwMndhnds 3-6

The expression of the strain can be rewritten as

em exeycxyT ;v cxzcyzvxvyvxyvyxT 3-7where

em e0m ze1m z3e3mv v0 z2v2 3-8

dU RXNx;xNxy;ydu0Ny;yNxy;xdv0 Nx1;xNxy1;y 12 Yyx;xYy;yQx cNx3;xNxy3;y 32

Ny1;yNxy1;x 12 Yxy;yYx;xQy cNy3;yNxy3;x 32

Qx;xQy;y 12 Yyx;xxYxy;yyYy;xyYx;xy

cNx3;xxNy3;yy2Nxy3;xy 32 Yyx2;xxYxy2;yyYy2;xyYx

26666664

R@X

mNxnNxydu0nNymNxydv0 12 Yyxm2Yxyn2dw;n 12Yxm2Ny1Nx1 12Yxy 12Yyxmn 12Yyn2Nxy1

dhn 12Yyx;xQx 14 Yx;yYy;y

m 12Yxy;yQy 14 Yx;xYy;x

32c Yy2n2Yx2m2 23Ny3Nx3Yyx2Yxy2

mn 23Nxy3

23 Nx3;xNxy3;yYyx2;x 12 Yx2;yYy2;y2Qx2

m 23Ny3

2Nx3Yyx2

m2 2Ny3Yxy2

n2

dw;n 2 Nx3N

26666666664

3 3 3w;xy hx;x;w;xy hy;y;w;yy hx;y;w;xx hy;xv2 c2xz c2yz v2x v2y v2xy v2yx

h iT 32c 2hy w;x;2hx w;y;

hx;x w;xy;hy;y w;xy;hx;y w;yy;hy;x w;xxT

3-9-2Substituting Eqs. (2-15), (3-8) and (3-9) into the Eq. (3-2), we

have,

dU Pnk1

dUk Pnk1

RVk r

kTm demmkTdvdxdydz

RX Pn

k1

R zk1zk

rkTm de0m zde1m z3de3m

mkT dv0 z2dv2 dz dxdy

RXNxNyNxy

8>:9>=>;

T

de0m Nx1Ny1Nxy1

8>:9>=>;

T

de1m Nx3Ny3Nxy3

8>:9>=>;

T

de3m

0B@

QxQyYxYyYxyYyx

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

9>>>>>>=>>>>>>;

T

dv0

Qx2Qy2Yx2Yy2Yxy2Yyx2

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

9>>>>>>>>=>>>>>>>>;

T

dv2

1CCCCCCCCCAdxdy

ZX

Nxdu0;xNydv0;yNxydu0;ydv0;xQx3cQx2dhyQy3cQy2dhxQx3cQx2dw;xQy3cQy2dw;y Nxy1cNxy3 12Yx 32cYx2

dhx;x Nxy1cNxy3 12Yy 32cYy2

dhy;y Nx1cNx3 12Yyx 32cYyx2

dhy;x Ny1cNy3 12Yxy 32cYxy2

dhx0y cNx3 12Yyx 32cYyx2

dw;xx 12Yxy 32cYxy2cNy3

dw;yy 12Yx3cYx2Yy3cYy2cNxy3

dw;xy

2666666664

3777777775dxdy

3-10As the expression given in Eq. (2-8), c 4l

3h2and l is a parameter

to control model, l = 0: rst order model, l = 1: the third orderReddy model.

By using twice the Greens theorem, we have

x2;xYy2;y3Qx2dhy

y2;yYx2;x3Qy2dhx

3Qx2;xQy2;ydw

37777775dxdyYxyYyxmn 14 YxYy

dw;t

Ny1 12Yxyn2 12 YxYymn Nx1 12Yyx

m2

dhs

dw

23Ny3Yxy2

n2Yy2Yx2mn 23Nx3Yyx2

m2

dhsNxy3;xYxy2;y 12 Yx2;xYy2;x2Qy2

ndw

Yxy2Yyx2mn 2Nxy3 1Yx2Yy2

dw;t

37777777775ds3 2

3-11

cn 3Qy2 y3@y xy3

@x4

y2

@x x2

@x2 xy2

@yV

where b .

truct12m2Yx12n

2Yymn Ny1Nx112Yxy12Yyx

Nxy132c n

2Yy2m2Yx2mn 23Ny3Nx3Yyx2Yxy2

Nxy3

Mn

m2Nx1n2Ny112m2Yyxn2YxymnYxYy

c n2Ny3m2Nx332mnYy2Yx232n2Yxy2m2Yyx2

Ms

12m2Yyxn2Yxyc n2Ny3m2Nx332m

2Yyx2n2Yxy2

014YxYy12mnYxyYyx c mn

32Yxy2Yyx2Nx3Ny3

3

where Nx, Ny, Nxy, Nxi, Nyi, Nxyi, Qx, Qy, Qx1, Qy1 are the classical trac-tions of the plate, Yx, Yxy, Yx2, Yy2, Yxy, Yyx, Yxy2, Yyx2 are the tractionof couple stress moment of the plate. They are

NaNa1Na3

8>>>>>>>>>>>>:3-17

where

Q k2 Qk44

Qk55

" #

eQ k2 eQ k44 eQ k55" #

bQ k2 eQ k44 eQ k55eQ k44 eQ k55" #

8>>>>>>>>>>>>>>>>>>>:3-18

2 2

ures 94 (2012) 21432156 2149Qy2

c0yz2

c2yz Qy22

c0yz2

c2yz3-19

4.2. Cross ply composite laminated Kirchhoff plate of new modiedcouple stress theory

Substituting geometric equations of hx @w@y ; hy @w@x and c = 0into (4-1), we have

Q11u0;xxQ66u0;yyQ12Q66v0;xy J11w;xxxJ122J66w;xyy fu0Q66Q12u0;xyQ66v0;xxQ22v0;yy J22w;yyy2J66 J12w;xxy fv 0@Qx@x

@Qy@y

2

2 eeQ 44 eeQ 55w;xxyy eeQ 55w;xxxx eeQ 44w;yyyy 12@fcy;x fcx;y fw0

8>>>>>>>>>>>>>>>

22 eeQ 55 eeQ 44w;xxyy eeQ 55w;xxxx eeQ 44w;yyyy 12 fcy;x @fcx;y

tructures 94 (2012) 21432156Yx

Yy

( )22Q2

v0x

v0y

8

Boundary conditions:

wjC 0@2w@x2

x0 or xL

0; @2w@y2

y0 or yL

0@hx@y

y0 or yL

0; @hy@x

x0 or xL

0

8>>>>>>>: 5-1A cross-ply simply supported laminated plate subjected to

bending loads of fw = q0sin(px/L)sin(py/L) is a simplest example.The traditional triangle function try method adopted in this paperand numerical result is only used to test the ability of the presenttheory and the model as well as the characteristics of the scale ef-fects of microstructure. It is of course that the numerical exampleis not used to study the advantages of the solving process.

5.1. Solution of the composite laminated Reddy plate with cross-ply ofcouple stress theory

Assume as u0(x,y) = 0, v0(x,y) = 0 and fu = fv = fcx = fcy = 0, and thetrial function is written as [28]

wx; y w0 sinax sin byhy hya cosax sinbyhx hxa sinax cosby

8>: 5-2

The stress at x = L/2, y = L/2 in section of the plate can be ex-pressed as follows:

rkx pL z pL cz2w0 Qk11 Qk12

1 cz2 Qk11hya Qk12hxa h i

;

rky pL z pL cz2w0 Qk12 Qk22

1 cz2 Qk12hya Qk22hxa h i

:

8>:5-5

where c 4l3h2

.

5.2. Numerical examples for the Scale effects of microstructure

In order to test characteristics of the scale effects of microstruc-ture, models of simply supported laminated cross-ply square plateare adopted. The sizes of the square plate model: length of a side isL = 200 lm, thickness is h = 25 lm, bending load is q0 = 1 N/lm2,the material constants [29]:E2 6:98 GPa; E1 25E2; G12 0:5E2; G22 0:2E2; m12 m22 0:25; vk21

Ek2vk12

Ek1, in which sub-

scripts 1 and 2 represent the direction of ber and matrix,respectively.

We choose the next two types of cross-ply laminated plate withthree-layer as rst one[0/90/0], and second one [90/0/90].

On the micro-materials constants2b ; 2m, for the composite lam-

inated plate, the curvatures of @xx@y and

@xy@x should be taken as inde-

pendent components which are relative to the micro materialsconstants 2b ;

2m to describe the materials microstructural charac-

teristics of the ber and matrix of the laminated plate respectively.

T22

12

12

12

T66

9

12

9

T66

W. Chen et al. / Composite Structures 94 (2012) 21432156 2151where a = p/L, b = p/L.Substituting Eq. (5-2) into equilibrium equations in terms of

displacements for the composite laminated Reddy plate, we have

l11w0 l12hya l13hxa q0 0l21w0 l22hya l23hxa 0l31w0 l32hya l33hxa 0

8>: 5-3where

l11 p2

L2Q44 Q55 6cI44 I55 9c2S55 S44 c

2p4

L4T11

l12 pL Q44 6cI44 9c2S44 p

3

L3cS11 S12 2S66 c2T11 T

l13 pL Q55 6cI55 9c2S55 p

3

L3cS22 S12 2S66 c2T22 T

l21 pL Q44 6cI44 9c2S44 p

3

L3cS11 S12 2S66 c2T11 T

l22 Q44 6cI44 9c2S44 p2

L2I11 I66 2cS11 S66 c2T11

l23 p2

L2I12 I66 2cS12 S66 c2T12 T66

2

4 eeQ 44 6ceeI 44

l31 pL Q55 6cI55 9c2S55 p

3

L3cS22 S12 2S66 c2T22 T

l32 p2

L2I12 I66 2cS12 S66 c2T12 T66

2

4 eeQ 55 6ceeI 55

l33 Q55 6cI55 9c2S55 p2

L2I22 I66 2cS22 S66 c2T22 In the couple stress theory, the rotation variables related to micro-scale impurities or defects are formulated into rotation equilibriumequations.

For kth layer of composite laminated plate in the local coordi-nates (x0,y0,z0), xx is a rotation round the ber and the its partialderivative @xx

@x ;@xx@y are dened as the curvatures vx, vxy, the

2b is

introduced constitute relations of the moments mx, mxy and thecurvatures vx, vxy. Thexy is a rotation to perpendicular the berand its partial derivative @xy

@x ;@xy@y are dened as the curvatures

2T12 4T66 2p4

2L4 eeQ 44 eeQ 55 6ceeI 44 eeI 55 9c2eeS44 eeS55

2T66 2p3

2L3 eeQ 55 9c2eeS55

2T66 2p3

2L3 eeQ 44 9c2eeS44

2T66 2p3

4L33 eeQ 55 4 eeQ 44 9c2eeS44 3eeS55

32p2

4L2 eeQ 55 6ceeI 55 9c2eeS55

c2eeS442T66

2p3

4L3 eeQ 55 3 eeQ 44 9c23eeS44 eeS55

c2eeS55 3

2p2

4L2 eeQ 44 6ceeI 44 9c2eeS445-4

vyx, vy. The 2m is introduced constitute relations of the momentsmyx, my and the curvatures vyx, vy. The micro-scale of impurities

with ber as the transverse isotropy in cross-section (y0,z0) is mushlarger than the micro-scale of matrix the transverse anisotropy incross-section (x0,z0). It is obviously, it is more easy to turn aroundber than turn to perpendicular to ber i.e.2b 2m, so for the lam-inated plate we can assume that 2m 0.

The values of the micro-materials constants 2b ; 2m, which re-

lated to the length in micro-scale of impurities or defects, in thenumerical example are considered in the scale on the thicknessof a single layer and 2m 0; 2b 2.

Next, keep the thickness of the plate constant and change thematerial constant l to examine the scale effect. Numerical resultsof the deection of the plate are given in Fig. 3 where, which showthat the deection of the plate in couple stress theory is smallerthan that in the classical elasticity (l = 0) as the material constantl increases.

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

Numerical results of the stresses in section of the plate are givenin Fig. 5, which show that the stress in the section of the plate incouple stress theory is smaller than that in the classical elasticityas the material constant l increases.

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

x

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

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

y

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 plate [0,90,0] at y = L/2.

2152 W.J. Chen et al. / Composite Structures 94 (2012) 214321560 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y/L

(a) at x L/2=-1.5 -1 -0.5 0 0.5 1 1.5-0.5

-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 stresses rx and ry in section o

Fig. 4. The angle of rotatio-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z/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 1-0.025

x/L(b) at /2y L=

n of the plate [0,90,0].y (GPa)

f the plate [0,90,0] at x = L/2, y = L/2.

xl=h

lat

truct0 0.1 0.2 0.3 0.40

0.005

0.01

0.015

0.02

0.025

0.03

w/h

Kirchhoff plateMindlin plateReddy plate

(a) Thick p

0.2

l=0l=h/2l=h

Kirchhoff plateMindlin plateReddy plate

W. Chen et al. / Composite SIt should be noted that the results of the second one [90/0/90] are equal to the rst one [0/90/0] by exchanging the x axisand y axis, for instance, the deection of the plate [90/0/90] atx = L/2 along y/L is same as the Fig. 3.

5.3. Solution of the cross-ply composite laminated Mindlin plate ofnew modied couple stress theory

Substituting u0(x,y) = 0, v0(x,y) = 0 and fu = fv = fcx = fcy = 0 intoEq. (4-1), the equilibrium equations in terms of displacement forthe composite laminated Mindlin plate of new modied couplestress theory can be expressed as follows:

Q44w;xx hy;x Q55w;yy hx;y 24 2eeQ 44 eeQ 55w;xxyy

eeQ 44w;yyyy 2hx;xxy hx;yyy eeQ 55w;xxxx 2hy;xyy hy;xxx fw 0Q44w;x hy I11hy;xx I66hy;yy I12 I66hx;xy

2eeQ 554 2wxyy w;xxx 2hy;yy hy;xx 0

Q55w;y hx I66hx;xx I22hx;yy I12 I66hy;xy

2eeQ 444 2w;xxy w;yyy 2hx;xx hx;yy 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:5-6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

x/L

w/h

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

(b) Medium thickness plate with / 8L h =Fig. 6. The deection of the plate to compare Kirchhoff plate, Min0.5 0.6 0.7 0.8 0.9 1

/L

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

e with / 4L h =

2500

l=0l=h/2l=h

Kirchhoff plateMindlin plateReddy platel=0l=h/2

ures 94 (2012) 21432156 2153Substituting the trial function (5-2) into Eq. (5-6), we have

3p424L4

eeQ 44 eeQ 55p2L2 Q44Q55

w0pL Q4432eeQ 55p24L2

!hya

pL Q5532eeQ 44p24L2

hxaq00

pL Q443

2eeQ 55p24L2

!w0 Q44p2L2

32eeQ 554 I11 I66

! !hyap2L2 I12 I66hxa0

pL Q553

2eeQ 44p24L2

!w0p2L2 I12 I66hya Q55p

2

L232eeQ 444 I22 I66

! !hxa0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:5-7

The stresses at x = L/2, y = L/2 in section of the plate can beexpressed as follows:

rkx z pL Qk11hya Qk12hxa

rky z pL Qk12hya Qk22hxa

8>: 5-8

5.4. Solution of the cross-ply composite laminated Kirchhoff plate ofcouple stress theory

Substituting u0(x,y) = 0, v0(x,y) = 0 and fu = fv = fcx = fcy = 0 intothe Eq. (4-5), the equilibrium equations in terms of displacement

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

x/L

w/h

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

(c) Thin plate with / 100L h =dlin plate and Reddy plate couple stress theories [0/90/0].

x (G

l=0

ate

truc-0.5 -0.4 -0.3 -0.2 -0.1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

z/h

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

(a) Thick pl

0.50.5

2154 W.J. Chen et al. / Composite Sof composite laminated Kirchhoff plate of cross-ply of couple stresstheory is given as

I11@4w@x4

2I12 2I66 @4w

@x2@y2 I22 @

4w@y4

2 eeQ 55 eeQ 44 @4w@x2@y2

eeQ 55 @4w@x4

eeQ 44 @4w@y4

! fw 5-9

The trial function is assumed as w(x,y) = w0sinaxsinby.The deection at x = L/2, y = L/2 in section of the plate can be

obtained as

w0 q0L4

p4 I11 2I12 2I66 I22 22 eeQ 44 eeQ 55 5-10The stresses at x = L/2, y = L/2 in section of the plate are

given as

rkx z Qk11 Qk12

pL

2w0rky z Qk12 Qk22

pL

2w08>: 5-11

-1.5 -1 -0.5 0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

x (GPa)

z/h

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

Kirchhoff plateMindlin plateReddy plate

(b) Medium thickness plate with / 8L h =Fig. 7. The stress rx in section of the plate at x = L/2 to compare Kirchhoff0 0.1 0.2 0.3 0.4 0.5

Pa)

Kirchhoff plateMindlin plateReddy plate

with / 4L h =

0.5

tures 94 (2012) 214321565.5. Numerical examples to compare Kirchhoff plate, Mindlin plate andReddy plate of new modied couple stress theories for microstructures

In order to compare Kirchhoff plate with Mindlin plate and Red-dy plate of the couple stress theories for microstructures, afore-mentioned models of simply supported cross-ply laminated plateare adopted. However, various sizes of the plate are chosen aslength L = 100 lm, L = 200 lm, L = 2500 lm and thickness h =25 lm respectively. We choose the cross-ply laminated plate withthree-layer of [0/90/0] and [90/0/90] respectively, and changethe material constant as l = (0,h/2,h), to examine the scale effect.

Numerical results of the deection of the plate are given inFig. 6 which show that the deference of the Mindlin plate and Red-dy plate in couple stress theory is more than Kirchhoff plate of cou-ple stress theory for thickness plate with L/h = 4, medium thicknessplate with L/h = 8, while the results for thin beam with L/h = 100are identical under the same as the material constant l.

It is should note that in the case of thin plate with L/h = 100, thelines of Reddy beam of couple stress theory coincide with Timo-shenko beam of couple stress theory and EulerBernoulli beamof couple stress theory in Figs. 6c, 7c, and 8c.

Numerical results of the stress rx in section of the plate are gi-ven in Figs. 7 and 8, which show that the stress rx of the Mindlinplate and Reddy plate in couple stress theory is smaller than

-250 -200 -150 -100 -50 0 50 100 150 200 250-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

x (GPa)

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

Kirchhoff plateMindlin plateReddy plate

(c) Thin plate with / 100L h =plate, Mindlin plate and Reddy plate couple stress theories [0/90/0].

y (Gte w

truct-0.4 -0.3 -0.2 -0.1-0.5

-0.4

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

(a) Thick pla

0.5l=0

W. Chen et al. / Composite SKirchhoff plate of couple stress theory for thickness beam with L/h = 4, Medium thickness beam with L/h = 8, while the results forthin beam with L/h = 100 are identical under the same as the mate-rial constant l.

6. Conclusions

A model for composite laminated Reddy plate of the new modi-ed couple stress theory is developed in rst time. The characteris-tics of the couple stress theory are the use of rotation-displacementas dependent variables and the use of only one constant to describethe materials microstructural characteristics. In this theory a newcurvature tensor is dened for establishing the constitutive rela-tions of laminated plate for anisotropy materials. The characteriza-tion of anisotropy is incorporated into higher-order laminated beamtheories based on themodied couple stress theory by Yang et al. in2002. The form of new curvature tensor is asymmetric, however itcan result in same as the symmetric curvature tensor in the caseof the isotropic elasticity. By introducing the hypothesis of thecross-section of plate, the governing equations of the compositelaminated Reddy plate of couple stress theory are established.

The present model of plate can be viewed as a simplied couplestress theory in engineering mechanics. A simply supported cross-ply laminated plate subjected to loads of fw = q0sin(px/L)sin(py/L)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

y (GPa)

z/h

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

Kirchhoff plateMindlin plateReddy plate

8(b) Medium thickness plate with /L h =Fig. 8. The stress rx in section of the plate at x = L/2 to compare Kirchhoff p0 0.1 0.2 0.3 0.4Pa)

Kirchhoff plateMindlin plateReddy plate

ith / 4L h =

0.5l=0ures 94 (2012) 21432156 2155is solved by directly applying the newly developed plate model.Numerical results show that the present plate model can capturethe scale effects of microstructure. The deections and stresses ofthe present model of plate of couple stress theory are always smal-ler than that by the classical plate model.

Additionally, the present Reddy plate model can be degeneratedto the model of cross-ply composite laminated Mindlin plate andcross-ply composite laminated Kirchhoff plate model of couplestress theory.

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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-80 -60 -40 -20 0 20 40 60 80-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

y (GPa)

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

Kirchhoff plateMindlin plateReddy plate

100 (c) Thin plate with /L h =late, Mindlin plate and Reddy plate couple stress theories [90/0/90].

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2156 W.J. Chen et al. / Composite Structures 94 (2012) 21432156

A model of composite laminated Reddy plate based on new modified couple stress theory1 Introduction1.1 Scale effects of microstructure1.2 Couple stress theory1.3 The couple stress isotropic plate model based on the C0 couple stress theory1.4 The couple stress isotropic plate model based on the C1 couple stress theory

2 Formulations for composite laminated plate based on new modified couple stress theory2.1 Classical couple stress theory2.2 Modified couple stress theory2.3 Hypothesis of composite laminated plate of new modified couple stress theory2.4 Two-dimensional plate theories2.5 Strain for modified couple stress theory of isotopic elasticity2.6 Constitutive relations for composite laminated plate based on new modified couple stress theory

3 Potential energy principle for composite laminated Reddy plate of new modified couple stress theory3.1 Equilibrium equations in terms of tractions for the composite laminated plate of new modified couple stress theory3.2 Equilibrium equations in terms of displacements for the composite cross-ply laminated plate of new modified couple stress theory

4 Degradation of the composite laminated plate of new modified couple stress theory4.1 Composite cross ply laminated Mindlin plate of new modified couple stress theory4.2 Cross ply composite laminated Kirchhoff plate of new modified couple stress theory

5 Numerical example for scale effect:simply supported plate5.1 Solution of the composite laminated Reddy plate with cross-ply of couple stress theory5.2 Numerical examples for the Scale effects of microstructure5.3 Solution of the cross-ply composite laminated Mindlin plate of new modified couple stress theory5.4 Solution of the cross-ply composite laminated Kirchhoff plate of couple stress theory5.5 Numerical examples to compare Kirchhoff plate, Mindlin plate and Reddy plate of new modified couple stress theories for microstructures

6 ConclusionsAcknowledgementReferences