Analytical solutions of heterogeneous rectangular plates with transverse small periodicity

Composites: Part B 43 (2012) 1056–1062

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Analytical solutions of heterogeneous rectangular plates with transverse small periodicity Wen-ming He a,b, Hua Qiao b,⇑, Wei-qiu Chen b a b

Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China Department of Engineering Mechanics, Yuquan Campus, Zhejiang University, Hangzhou 310027, PR China

a r t i c l e

i n f o

Article history: Received 29 June 2011 Received in revised form 21 August 2011 Accepted 14 September 2011 Available online 28 September 2011 Keywords: A. Laminates A. Plates B. Microstructures C. Analytical modelling Periodic structures

a b s t r a c t We consider the cylindrical bending of a simply-supported orthotropic rectangular plate, which is smallperiodically heterogeneous in the thickness direction. A homogenized plate model is first established by using the two-scale asymptotic expansion method. The state-space method is then adopted to analyze the homogenized plate exactly. Analytical expressions for two sets of approximate stresses, i.e. the homogenized model stresses and the zeroth-order two-scale model stresses, are presented. To check the accuracy of the approximate stresses, the state-space method is also applied to the original laminate plate or the approximate laminate model of the original heterogeneous plate with continuously varying material properties to obtain an analytical solution. In the latter case, the analytical solution is approximate but approaches the exact solution gradually when the number of layers increases. In order to avoid numerical instability, the joint coupling matrix is utilized. Numerical results illustrate that the two-scale model can predict accurately the realistic stress field in the original plate if it contains enough repeated units along the thickness. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Heterogeneous structures, including laminated composite structures and functionally graded material structures as two particular cases, have found wide applications in various engineering applications such as aerospace, ocean, mechanical, and civil, due to their excellent performance. In practice, it is very important to know exactly the responses of heterogeneous structures when they are subjected to static or dynamic excitations. A lot of analytical and numerical methods thus have been developed. One typical catalogue of the methods is the simplifying theories, which reduce the three-dimensional (3D) governing equations to a set of twodimensional (2D) differential equations for plates and shells or even to a set of one-dimensional (1D) differential equations for beams, by assuming proper and usually known distribution functions for physical quantities along the thickness direction [11,24]. Because of the dimension reduction, the problems formulated in the framework of simplifying theories are much easier to deal with. Another catalogue is the exact analyses starting directly from 3D or 2D elasticity theory, which however work only for few cases, such as simply-supported, cross-ply laminated plates. The two typical exact solutions obtained by Pagano [20,21] are well-known and have been frequently used as benchmarks for clarifying various ⇑ Corresponding author. Tel./fax: +86 571 87952733. E-mail address: [email protected] (H. Qiao). 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2011.09.010

simplifying analyses or numerical simulations. In recent decades, exact analyses based on state-space formulations have proven to be powerful and effective in the study of laminated composite structures [7–10,16]. The propagator matrix method has also been developed to analyze mechanical behavior of laminated composite structures [18,19]; it is basically equivalent to the state-space approach, with however an explicit expression presented for the inverse of the eigenvector matrix. In addition to the above two catalogues, multiscale asymptotic methods have also been proposed to study the behavior of composite materials and structures with small periodic structures [1,2,15,25], and proved to be very efficient in a variety of applications [4,3,12]. By such methods, it becomes possible to predict both the overall and local responses of the composites. It is interesting to note that, when dealing with laminated structures, the multiscale asymptotic methods have been mainly used to obtain the effective moduli [12]. In this paper, the two-scale asymptotic method of homogenization is first employed to establish a homogenized model of a rectangular composite plate with a small periodic structure in the thickness direction. Then, the state-space method is used to obtain the exact response of the homogenized plate under external vertical loads. Two sets of approximate stresses (the stresses of the homogenized model and the zeroth-order stresses of the two-scale asymptotic expansion model) are then presented analytically. The accuracy of these approximate stresses is numerically checked by

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comparing with an exact or highly accurate approximate solution of the original plate, which is also obtained using the state-space formulations, along with the joint coupling matrix to ensure numerical stability.

In view of these expressions, we obtain from Eq. (2),

2. Mathematical description of the problem

r0xx ¼ c11 ð#Þ

Shown in Fig. 1 is a rectangular plate of length a and thickness h, which deforms in a cylindrical bending state so that u ¼ uðx; yÞ, v ¼ v ðx; yÞ and w ¼ 0. Here u, v and w are the displacement components in the x, y and z-directions, respectively. Such a state can be achieved by assuming that the plate is infinite along the z-axis and the external loads and geometric constraints do not vary with z. For cylindrical bending deformation, the equations of equilibrium are

@ rxx @ rxy @ rxy @ ryy þ ¼ 0; þ ¼ 0: @x @y @x @y

ð1Þ

rij ¼ r0ij ðx; y; #Þ þ er1ij ðx; y; #Þ þ    ði; j ¼ x; yÞ;

ð6Þ

where

r0yy r0xy

 0  @u0 @v 1 @v 1 þ c12 ð#Þ þ ; @x @y h @#   @u0 @v 0 1 @v 1 ; ¼ c12 ð#Þ þ c22 ð#Þ þ @x @y h @#  0  @u 1 @u1 @v 0 ¼ c66 ð#Þ þ þ c66 ð#Þ : @y h @# @x

ð7Þ

 1  @u1 @v 1 @v 2 þ c12 ð#Þ þ ; @x @y h @#   @u1 @v 1 1 @v 2 ¼ c12 ð#Þ þ c22 ð#Þ þ ; @x @y h @#  1  @u 1 @u2 @v 1 ¼ c66 ð#Þ þ þ c66 ð#Þ : @y h @# @x

ð8Þ

r1xx ¼ c11 ð#Þ r1yy r1xy

where rij denote the stress components. We confine ourselves to orthotropic materials, of which the constitutive relations are expressed as   @u @v @u @v @u @ v þ rxx ¼ c11 þ c12 ; ryy ¼ c12 þ c22 ; rxy ¼ c66 ; ð2Þ @x @x @y @x @y @y

Substituting Eq. (6) into the equilibrium equations, Eq. (1), and equating the coefficients of e1 and e0, we obtain

where cij are the elastic constants. In this paper, we assume that the plate has a small periodic structure along the y-direction, i.e. it is heterogeneous and cij depend on the coordinate y. Thus, both laminated and functionally graded cases are included in this model. The plate is simply-supported at the two edges x ¼ 0; a:

@ r0xx @ r0xy 1 @ r1xy þ þ ¼ 0; h @# @x @y

rxx ¼ v ¼ 0 at x ¼ 0; a:

ð3Þ

The boundary conditions at the bottom and top surfaces y ¼ 0; h will be discussed later.

@ r0xy ¼ 0; @#

@ r0yy ¼ 0: @#

 @ 1 c66 ð#Þ @# h  @ 1 c22 ð#Þ @# h

@u1 @#



@v 1 @#

¼

dc66 ð#Þ @u0 dc66 ð#Þ @ v 0  ; d# d# @y @x

¼

dc22 ð#Þ @ v 0 dc12 ð#Þ @u0  ; d# d# @y @x



If we assume

The multiscale asymptotic methods of homogenization have become a powerful tool to predict both the overall and local properties of processes in composites. Detailed description of multiscale asymptotic methods may be referred to, for instances, SanchezPalencia [25], Bakhvalov and Panasenko [3], and Kalamkarov [12]. For the heterogeneous plate shown in Fig. 1, we introduce the local dimensionless coordinate # by

u1 ðx; y; #Þ ¼ hN uu ð#Þ

ð4Þ

where e is a small dimensionless parameter, representing the size ratio between the unit cell and the thickness of the whole plate, h. It is clear that cij are eh-periodic in the y-direction, while cij ðy=ðehÞÞ ¼ cij ð#Þ is 1-periodic in view of Eq. (4). Now following the two-scale expansion method of homogenization (see e.g. [3], we write the displacements as follows

u ¼ u0 ðx;yÞ þ eu1 ðx;y;#Þ þ ;

v ¼ v 0 ðx;yÞ þ ev 1 ðx;y;#Þ þ ;

ð5Þ

@ r0xy @ r0yy 1 @ r1yy þ þ ¼ 0: h @# @x @y

ð10Þ

Eq. (7) is now substituted into Eq. (9) to give

3. Two-scale asymptotic method of homogenization

# ¼ y=ðehÞ

ð9Þ

@u0 @v 0 þ hN uv ð#Þ ; @y @x @u0 @v 0 v 1 ðx; y; #Þ ¼ hNv u ð#Þ þ hNvv ð#Þ ; @x @y

ð11Þ

ð12Þ

where N ij ð#Þði; j ¼ u; v Þ are 1-periodic in #, i.e. N ij ð0Þ ¼ N ij ð1Þ, then we can get from Eq. (11)

  d dNuu ð#Þ dc66 ð#Þ c66 ð#Þ ¼ ; d# d# d#   d dNuv ð#Þ dc66 ð#Þ c66 ð#Þ ¼ ; d# d# d#   d dNv u ð#Þ dc12 ð#Þ c22 ð#Þ ¼ ; d# d# d#   d dNvv ð#Þ dc22 ð#Þ ¼ c22 ð#Þ : d# d# d#

ð13Þ

which further lead to

a

y z

εh h

dNuu ð#Þ A1 dNuv ð#Þ A2 ¼ 1 þ ¼ 1 þ ; ; d# d# c66 ð#Þ c66 ð#Þ dNv u ð#Þ c12 ð#Þ A3 dN vv ð#Þ A4 ¼ ¼ 1 þ þ ; ; d# c22 ð#Þ c22 ð#Þ d# c22 ð#Þ

where Ai(i = 1, 2, 3, 4) are integral constants. Since Nij ð#Þ ði; j ¼ u; v Þ are 1-periodic functions, we can by integrating Eq. (14) over # 2 ½0; 1 get the following relations:

x

A1 ¼ A2 ¼ Fig. 1. A composite rectangular plate with a small periodic structure in the thickness direction.

ð14Þ

1 ; hc1 66 i

A3 ¼

hc1 22 c 12 i ; hc1 22 i

A4 ¼

1 hc1 22 i

ð15Þ

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R1

where hcij i ¼ 0 cij ð#Þd# are the macroscopic (or average) material constants. Then, Eq. (14) can be written as

4. Exact analysis of the homogenized plate

dNuu ð#Þ dNuv ð#Þ 1 ¼ ¼ 1 þ ; d# d# c66 hc1 66 i

Although there are different methods that can be used to derive the exact solution for the simply-supported homogenized plate subjected to transverse loads, we will adopt here the state-space method which has recently attracted much attention from researchers [10,13,9]. Moreover, we will present in the next section the solution of the original plate for which, the state-space method becomes particularly very effective. For deformation of the homogenized plate governed by Eqs. (17) and (18), the following state equation can be derived:

dNv u ð#Þ c12 ð#Þ hc1 22 c 12 i ¼ þ ; d# c22 ð#Þ c22 ð#Þhc1 22 i dNvv ð#Þ 1 ¼ 1 þ : d# c22 ð#Þhc1 22 i

ð16Þ

We now proceed with the establishment of the homogenized model of the original heterogeneous plate. Integrating Eq. (10) over # 2 ½0; 1 and noticing that r1ij ðx; y; #Þ are 1-periodic in #, we get

@hr0xx i @hr0xy i þ ¼ 0; @x @y

@hr0xy i @hr0yy i þ ¼0 @x @y

ð17Þ

where hr0ij i are the macroscopic stresses defined by R1 hr0ij i ¼ 0 r0ij ðx; y; #Þd#. Eq. (17) gives the governing equations for the homogenized model. The macroscopic stresses are related to the zeroth-order displacements by

@u0 ðx; yÞ @ v 0 ðx; yÞ þ C 12 ; hr0xx i ¼ C 11 @x @y @u0 ðx; yÞ @ v 0 ðx; yÞ hr0yy i ¼ C 12 þ C 22 ; @x @y  0  @u ðx; yÞ @ v 0 ðx; yÞ hr0xy i ¼ C 66 þ : @y @x

@ 0 ½u ; hr0yy i; hr0xy i; v 0 T ¼ M½u0 ; hr0yy i; hr0xy i; v 0 T ; @y

where u0 , v 0 , hr0yy i and hr0xy i are called state variables, the superscript T indicates matrix transpose, and

2

1 ; hc1 22 i

C 12 ¼

2 hc1 22 c 12 i ; 1 hc22 i

ð18Þ

C 66 ¼

hr0xx i ¼ v 0 ¼ 0 at x ¼ 0; a;

1 : hc1 66 i

ð19Þ

ð20Þ

The surface boundary conditions at y ¼ 0; h can be similarly derived and will be given in Section 4. With the homogenized model, the zeroth-order stresses r0ij in the two-scale asymptotic expansion can be obtained according to Eqs. (7), (12), and (16) as follows   0 c2 ð#Þ c ð#Þhc1 c12 ð#Þ @ v 0 ðx; yÞ 22 c 12 i @u ðx; yÞ þ ; r0xx ¼ c11 ð#Þ  12 þ 12 1 c22 ð#Þ @x @y c22 ð#Þhc22 i c22 ð#Þhc1 22 i

hc1 c i @u0 ðx;yÞ 1 @ v 0 ðx;yÞ þ 1 ; r ¼ 22112 @x @y hc22 i hc22 i  0  1 @u ðx;yÞ @ v 0 ðx; yÞ þ ; r0xy ¼ 1 @y @x hc66 i 0 yy

ð21Þ

Thus, we have two sets of approximate stresses, the homogenized model stresses (or macroscopic stresses) hr0ij i and the zeroth-order two-scale model stresses r0ij , given by Eqs. (18) and (21), respectively. The relation between them can be easily found as

  c12 0 0  jhc12 c1 22 i hryy i þ jhrxx i; c22

r0xy ¼ hr0xy i; r0yy ¼ hr0yy i; ð22Þ

where

j¼

c11  c212 c1 22 : hc11 i  hc212 c1 22 i

1 C 66

0

@  @x

 CC 12 22

@ @x

@  @x

7 0 7 7 7: 0 7 7 5 0

0

1 C 22

3

0

ð25Þ

hr

0 xx i

¼

! C 212 @u0 ðx; yÞ C 12 0 þ C 11  hr i: @x C 22 C 22 yy

ð26Þ

Since the plate is simply-supported at x ¼ 0; a and for the homogenized model we have the boundary conditions as shown in Eq. (20), we may assume

Eq. (18), derived from Eqs. (7), (12), and (16), represents the macroscopic constitutive relations of the homogenized model. The boundary conditions at the two edges for the homogenized plate can be derived from that for the original plate, i.e. Eq. (3), and the two-scale expansions in Eqs. (5) and (6). The results are

r0xx ¼

0

The so-called induced variable hr0xx i can be simply calculated from the state variables as

hc1 22 c 12 i ; hc1 22 i

C 11 ¼ hc11 i  hc212 c1 22 i þ

0

6 6 0 6  2 2 M¼6 C 6 12  C @ 11 @x2 6 C 22 4 @  CC 12 22 @x

where

C 22 ¼

ð24Þ

ð23Þ

8 0 9 u > > > > > > > < hr0 i > = yy

> hr > > > : 0 > ; 0 > xy i > >

2 ¼

3

hUðgÞ cosðmpnÞ

1 6 7 X 6 C 11 Ryy ðgÞ sinðmpnÞ 7 7; 6 4 C 11 Rxy ðgÞ cosðmpnÞ 5 m¼1

ð27Þ

hVðgÞ sinðmpnÞ

v

where n ¼ x=a and g ¼ y=h are the dimensionless coordinates, and m is an integer. Making use of Eq. (27), we obtain from Eq. (24)

dVðgÞ ¼ MVðgÞ dg

ð28Þ

for each m, where VðgÞ ¼ ½UðgÞ; Ryy ðgÞ; Rxy ðgÞ; VðgÞT is the dimensionless state vector, and

2

0

6 6 0  6 M ¼ 6 C 212 6 C 22 C 11  1 t 2 4 C 12 t C 22

0

 CC 11 66

0

t

C 12 C 22

t

C 11 C 22

0 0

t

3

7 0 7 7 7 0 7 5 0

ð29Þ

with t ¼ mph=a. According to the matrix theory, we can write the solution to Eq. (28) as

VðgÞ ¼ expðMgÞVð0Þ;

ð30Þ

Setting g ¼ 1 in Eq. (30) leads to a transfer relation between the state vector at the upper surface V1 and that at the lower surface V0 as

V1 ¼ TV0 ;

ð31Þ

where T ¼ expðMÞ is the transfer matrix of the homogenized plate. Now let’s consider the bending deformation of the plate. Assume that generally distributed normal tractions pðxÞ and qðxÞ are applied on the bottom and top surfaces, respectively. Thus, we have the following surface boundary conditions:

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W.-m. He et al. / Composites: Part B 43 (2012) 1056–1062

Ryy ð0Þ ¼ am ;

Ryy ð1Þ ¼ bm ;

Rxy ð0Þ ¼ Rxy ð1Þ ¼ 0

ð32Þ

for each m, where

½am ; bm  ¼

2 C 11

Z

1

½pðnaÞ; qðnaÞ sinðmpnÞdn

ð33Þ

ðkÞ V0

where and are the state vectors at the upper and lower sur g  g Þ is the faces, respectively, of the kth layer, and Tk ¼ exp½Mð k k1 transfer matrix of that layer. Similarly, one gets

0

8 8 9 9 Uð1Þ > Uð0Þ > > > > > > > > > > > < < = bm am = ¼T ; > > > 0 > > 0 > > > > > > > : : ; ; Uð1Þ Vð0Þ

ðkþ1Þ

V1

ð34Þ

ðkþ1Þ V0

ðNÞ

ð1Þ

V1 ¼ TV0 :

ð40Þ

QN

To check the accuracy of the two sets of approximate stresses derived from the homogenized plate, we need to know the true stresses in the original heterogeneous plate. In this section, we will employ the state-space formulations to derive an exact solution for the original plate if its material parameters are piece-wise constant or an approximate solution if its material parameters vary continuously along the thickness. In the latter case, an approximate laminate model will be employed to transform the original plate to a laminated plate in which each individual layer is homogeneous with its material parameters equal to those at its middle surface. It is obvious that the approximate solution can be made very accurate by increasing the layer number of the laminate model. Now let us consider an N-layered laminate plate, which may be either the original plate or its approximate model as mentioned above. The state equation (24) is still valid for each layer in the laminate except that the displacements u0 ; v 0 , stresses hr0yy i; hr0xy i, and elastic constants C ij should be replaced by u; v , ryy ; rxy , and cij , respectively. Similarly, we can assume

 ðgÞ cosðmpnÞ hu

7 ð1Þ 1 6 X 6 c11 r yy ðgÞ sinðmpnÞ 7 7; 6 ¼ 6 ð1Þ > rxy >  xy ðgÞ cosðmpnÞ 7 > > 5 m¼1 4 c11 r > > : ; v hv ðgÞ sinðmpnÞ

ð35Þ

VðgÞ ¼ exp½Mðg  gk1 ÞVðgk1 Þ ðgk1 6 g 6 gk ; k ¼ 1; 2; . . . ; NÞ ð36Þ  ðgÞ; r  yy ðgÞ; r  xy ðgÞ; v  ðgÞT , g0 ¼ 0, gk ¼ yk =h ¼ where VðgÞ ¼ ½u Pk j¼1 hj =h and hk is the thickness of the kth layer. The coefficient matrix M now reads

0

 c66

0

t

c12 c22

t

0

ð1Þ

c11 c22

0

( Jm

t

3

7 0 7 7 7 7; 0 7 7 5 0

)

ðkÞ

V1

¼0

Vkþ1 0

ð41Þ

where Jm ¼ ½I44  I44  is the joint coupling matrix, and I is a unit matrix. The surface boundary conditions also can be written as ð1Þ

ðNÞ

Jb V0 ¼ f b ;

Jt V1 ¼ f t ;

ð42Þ

where Jb and Jt are, respectively, the joint coupling matrices at the bottom and top surfaces of the plate:

Jb ¼ Jt ¼



0 1 0 0



0 0 1 0

;

ð43Þ

and f b and f t are given by

fb ¼



am ; 0

ft ¼



bm



0

ð44Þ

where am and bm are still given by Eq. (33) but with C 11 replaced ð1Þ with c11 . Thus, we can derive the following global relations for all state vectors at each interface/surface from Eqs. (41) and (42)

JV G ¼ f

ð1Þ

where c11 represents the elastic constant of the first layer (i.e. the bottom layer) in the laminate. We then get an equation similar to Eq. (28), whose solution for each individual layer can be written as

ð1Þ c11

where T ¼ j¼1 Tj is the global transfer matrix. Theoretically, we then can obtain the complete solution in the same way as described in the end of last section. However, we have encountered serious numerical difficulty using such a formulation in the computation, which is actually an intrinsic problem of the state-space method or the transfer matrix method (see for example, [23,22]. Thus, we further employ the concept of joint coupling matrix which was proposed by Nagem and Williams [17] to avoid the above-mentioned difficulty. To this end, the continuity conditions are rewritten in the following form:

3

yy

0 6 6 0  6 6 M ¼ 6 c212 c11 t2 6 c cð1Þ  cð1Þ 6 22 11 11 4 c12 t c22

ð39Þ

:

Making use of the continuity conditions ¼ at each interface between two adjacent layers, we could arrive at a global transfer relation as

5. Accurate analysis of the original plate

2

ðkþ1Þ

¼ Tkþ1 V0

ðkÞ V1

from which the four unknown state variables at the upper and bottom surfaces can be determined and those at any interior point can be calculated according to Eq. (30). The zero-order stresses are then determined from Eq. (21) or (22).

2

ð38Þ

ðkÞ V1

are the Fourier series coefficients of respective loads. With Eq. (32), the transfer relation in Eq. (31) becomes

8 9 u > > > > > =

ðkÞ

Vk1 ¼ Tk V0

ð45Þ

where

2

3

J ¼ diag4Jb Jm    Jm Jt 5; |fflfflfflffl{zfflfflfflffl} N1

h iT ð1ÞT ð1ÞT ð2ÞT ð2ÞT ðNÞT ðNÞT V G ¼ V 0 V1 V0 V 1    V 0 V1 ; 2 3T T

ð46Þ

T

T f ¼ 4f b 0   0ffl}T f t 5 : |fflfflfflfflffl{zfflfflfflffl N1

Eq. (38) can also be written as

ð37Þ

in which cij , although assumed constant in each individual layer, vary from layer to layer. Setting g ¼ gk in Eq. (36), gives

(

ðkÞ

V0

ðkÞ

V1

)

 ¼

I Tk



ðkÞ

V0 :

ð47Þ

With this, we can rewrite Eq. (45) as

JQ VG0 ¼ f; where

ð48Þ

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ð1ÞT

I T1



ð2ÞT

VG0 ¼ ½V0 V0

I T2



 

I TN

 ;

ðNÞT T

   V0

 :

Then we obtain the state vectors at the lower surfaces of each individual layer as

VG0 ¼ ðJQ Þ1 f:

.52

ð49Þ

ð50Þ

The complete solution thus can be determined. 6. Numerical results and discussions

Normal stress in y direction at x=a/2

Q ¼ diag



As the first example, we consider a simply-supported rectangular plate with length a ¼ 1 and thickness h ¼ 0:5. The small parameter is taken to be e ¼ 0:1, indicating that there are totally 10 repeated units in the thickness direction. The loads applied on the top and bottom surfaces of the plate are given by:

.44 .40 .36 .32 .28 .24 0.0

In addition, the elastic constants, which are eh-periodic in the y-direction, are assumed to be:

-.02

rij  r0ij ¼ er1ij þ     er1ij ;

ð53Þ

.3

.4

.5

Original model Homogenized model

Shear stress at x=a/4

-.04 -.06 -.08 -.10 -.12 -.14 -.16 -.18

0.0

.1

.2

.3

.4

.5

y=[0~h]

(b) Shear stress 1.0

Normal stress in x direction at x=a/2

We display in Fig. 2 the stresses calculated based on the homogenized model, the zeroth-order two-scale model and the original model. The normal stresses are obtained for n ¼ 0:5 (i.e. the midspan of the plate), while the shear stresses are for n ¼ 0:25. It is seen that the homogenized model gives almost exactly the same normal stress in the y-direction and the same shear stress as the original heterogeneous plate. However, the homogenized model seems not able to predict the normal stress in the x-direction. In contrast, all stress components calculated from the zeroth-order two-scale model agree perfectly with those of the original model. In the second example, the plate length, the elastic moduli, the small parameter e, and the loads are the same as those of the first example, except for the thickness h ¼ 0:1. Thus, the thickness of the plate in this example is much smaller than that of the plate in the first example. The results are given in Fig. 3, from which one can make the same conclusion as that from the first example. It indicates that the accuracy of the two-scale model does not depend directly on the thickness of the plate. In the third example, all the parameters are taken to be the same as the second example, except for e ¼ 0:5, which means that there are only two unit cells in the thickness direction. The calculated stresses are given in Fig. 4, showing that the shear stress and the normal stress in the y-direction as predicted by the homogenized model become less accurate than those in the last two examples, and the normal stress in the x-direction of the zeroth-order two-scale model also deviate a bit from that of original model. Thus, from the above three numerical examples, we may conclude that the accuracy of stresses calculated from the two-scale model mainly depends on the small parameter e, as evidenced by the two-scale expansions in Eqs. (5) and (6). We take a further step to check the error between the real stresses rij and the zeroth-order two-scale model stresses r0ij . In fact, when e ! 0, we get from Eq. (6)

.2

(a) Normal stress in the y-direction 0.00

  1 2py 1 c11 ðx; yÞ ¼ c22 ðx; yÞ ¼ 4 þ sin ¼ 4 þ sinð2p#Þ; 2 2 eh   1 2py 1 ¼ 2 þ cosð2p#Þ; c12 ðx; yÞ ¼ 2 þ cos 2 2 eh 1 1 c66 ðx; yÞ ¼ ½c11 ðx; yÞ  c12 ðx; yÞ ¼ 1  ½sinð2p#Þ  cosð2p#Þ: 2 4 ð52Þ

.1

y=[0~h]

ð51Þ

qðxÞ ¼ xð1  xÞ pðxÞ ¼ 2xð1  xÞ:

Original model Homogenized model

.48

Original model Homogenized model Two-scale model (zeroth-order value)

.8 .6 .4 .2 0.0 -.2 -.4 -.6 -.8

0.0

.1

.2

.3

.4

.5

y=[0~h]

(c) Normal stress in the x-direction Fig. 2. Comparison of stresses for example 1.

where r1ij =C 11 will depend on the aspect ratio of the plate (i.e. h=a), the stiffness ratios, as well as the dimensionless loads, as one may see from Section 4, where the dimensional analysis for the macroscopic stresses (hr0ij i) should be also valid for the first-order twoscale model stresses (r1ij ). Thus for small values of e, the relationship between jrij  r0ij j and e is approximately linear. Assume that the plate length, thickness, the elastic moduli, and the loads are all

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.52

Normal stress in y direction at x=a/2

Normal stress in y direction at x=a/2

.52 Original model Homogenized model

.48 .44 .40 .36 .32 .28 .24 0.00

.02

.04

.06

.08

.44 .40 .36 .32 .28 .24 0.00

.10

Original model Homogenized model

.48

.02

.04

0.0 Original model Homogenized model

Original model Homogenized model

Shear stress at x=a/4

-.2

-.4

-.6

-.4

-.6

-.8

-.8

.02

.04

.06

.08

-1.0 0.00

.10

.02

.04

.06

y=[0~h]

y=[0~h]

(b) Shear stress

(b) Shear stress

.08

.10

20

20

Original model Homogenized model Two-scale model (zeroth-order value)

16 12

Normal stress in x direction at x=a/2

Shear stress at x=a/4 Normal stress in x direction at x=a/2

.10

0.0

-.2

8 4 0 -4 -8 -12 -16 -20 0.00

.08

(a) Normal stress in the y-direction

(a) Normal stress in the y-direction

-1.0 0.00

.06

y=[0~h]

y=[0~h]

Original model Homogenized model Two-scale model (zeroth-order value)

16 12 8 4 0 -4 -8 -12 -16

.02

.04

.06

.08

.10

y=[0~h]

(c) Normal stress in the x-direction Fig. 3. Comparison of stresses for example 2.

the same as the second example. The numerical results in Fig. 5 confirm the theoretical prediction from Eq. (53). 7. Summary In this paper, we combine the two-scale asymptotic expansion technique and the state-space approach to develop an analytical

0.00

.02

.04

.06

.08

.10

y=[0~h]

(c) Normal stress in the x-direction Fig. 4. Comparison of stresses for example 3.

solution for a simply-supported composite plate with a small periodic structure in the thickness direction. The two-scale asymptotic expansion technique is used to obtain the homogenized model of the original heterogeneous plate which is solved by the state-space approach. Numerical examples are considered to clarify the effectiveness of the analytical method. Results show that the zeroth-order stresses of the two-scale model agree well with the original problem

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References

7 Normal stress (σ xx − σ xx0 ) Normal stress (σ yy − σ yy0 ) Shear stress ( σ xy − σ xy0 )

Relative error (%)

6 5 4 3 2 1 0 0.0

.1

.2

.3

.4

.5

Small parameter ε Fig. 5. Relative error of the zeroth-order stress.

provided that the number of cell units along the thickness is large enough. Further, it is verified that the relationship between the error of the zeroth-order stresses of the two-scale model and e is approximately linear, which is coincident with the theory. It can be concluded that the developed analytical solution can be a benchmark for the analysis of complex composite structures with small periodic structures. It is noted that this paper only considers the case of simple supports at the two ends of the plate, which allows analytical solutions both for the original plate or the homogenized plate. If other boundary conditions are considered, then we may combine the state-space approach with certain numerical schemes (e.g. the state-space based differential quadrature method, see Chen et al. [5,6] for example), or in a completely different framework formulated in a symplectic space [26,14]. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 11090333, 10725210, 11171257, and 90916027), the Special Funds for Major State Basic Research Projects (973 Program, No. 2010CB832702), the China Postdoctoral Science Foundation (No. 20090451454), the Zhejiang Provincial Natural Science Foundation, China (No. Y6090108), and the Fundamental Research Funds for the Central Universities.

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