Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure

Composite Structures 93 (2011) 722–735

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Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure Sh. Hosseini-Hashemi ⇑, M. Fadaee, S.R. Atashipour School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran

a r t i c l e

i n f o

Article history: Available online 10 August 2010 Keywords: Exact closed-form solution Free vibration Functionally graded rectangular plates Third-order shear deformation plate theory

a b s t r a c t In this article, a new exact closed-form procedure is presented to solve free vibration analysis of functionally graded rectangular thick plates based on the Reddy’s third-order shear deformation plate theory while the plate has two opposite edges simply supported (i.e., Lévy-type rectangular plates). The elasticity modulus and mass density of the plate are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents whereas Poisson’s ratio is constant. Based on the present solution, five governing complicated partial differential equations of motion were exactly solved by introducing the auxiliary and potential functions and using the method of separation of variables. The validity and high accuracy of the present solutions are investigated by comparing some of the present results with their counterparts reported in literature and the 3-D finite element analysis. It is obvious that the present exact solution can accurately predict not only the out of plane, but also the in-plane modes of FG plate. Furthermore, a new eigenfrequency parameter is defined having its special own characteristics. Finally, the effects of boundary conditions, thickness to length ratio, aspect ratio and the power law index on the frequency parameter of the plate are presented. Published by Elsevier Ltd.

1. Introduction In recent years, functionally graded materials (FGMs) have gained such popularity as high thermal resistance materials with low thermal stresses that structural components exposed to high-temperature environments such as aircraft structures are made of FGMs. They are a new generation of composite structures first introduced by a group of Japanese scientists in 1984 [1,2]. Typically, FGMs are made of a ceramic and a metal in such a way that the ceramic can resist the severe thermal loading from the hightemperature environment, whereas the metal is served in order to decrease the large tensile stress occurring on the ceramic surface at the earlier stage of cooling. Since the material properties of FGMs vary continuously from one interface to the other, this results in eliminating interface problems of composite materials and achieving the smooth stress distribution. Rectangular thick plates made of FGMs are often employed as a part of engineering structures. Analytical studies on FG rectangular thick plates are, however, rare whereas several studies have been numerically performed to analyze the mechanical, thermal or the thermo-mechanical responses of FG rectangular plates. When the plate is thin, the classical plate theory (CPT) is used to analyze FG rectangular plate problem. In an outstanding work on * Corresponding author. Tel.: +98 21 7391 2912; fax: +98 21 7724 0488. E-mail address: [email protected] (Sh. Hosseini-Hashemi). 0263-8223/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.compstruct.2010.08.007

the free vibration analysis of thin rectangular plates with a pair of opposite edges simply supported, exact solutions were presented by Leissa [3] for all possible combinations of classical boundary conditions along the other edges. Free vibration of FG simply-supported and clamped rectangular thin plates was considered by Abrate [4] using the CPT. Woo et al. [5] provided an analytical solution for the nonlinear free vibration behavior of FG square thin plates using the von Karman theory. Based on the CPT and an analytical method, Yang and Shen [6] presented free vibration and transient response of initially stressed FG rectangular thin plates subjected to impulsive lateral loads, resting on Pasternak elastic foundation. Also, Shen [7] and Yang and Shen [8] studied large deflections and postbuckling response of functionally graded plates with temperature-dependent material properties using a classical plate theory and a perturbation method. Javaheri and Eslami [9] studied the thermal buckling of the FG plate using the analytical solution based on the classical plate theory. Ng et al. [10,11] presented the first contribution to the dynamic stability investigations of FGM structures based on the CPT, without considering thermal environment and aerodynamic load. Furthermore, the governing equations were obtained using Hamilton’s principle and the solutions were analytically determined by employing the assumed mode technique. Due to ignoring the effect of shear deformation through the plate thickness, the CPT is valid only for thin plates and gives proper results for lower frequencies. In addition, the CPT underestimates

Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

deflections and overestimates frequencies. In order to eliminate the deficiency of the CPT for moderately thick plates, the first-order shear deformation theory (FSDT), including the effects of transverse shear deformation and rotary inertia, was employed by many research groups using analytical and numerical methods. A very beneficial study for deriving the exact closed-form characteristic equations of freely vibrating isotropic rectangular Mindlin plates for the six cases having two opposite sides simply supported was carried out by Hosseini-Hashemi and Arsanjani [12]. A similar study was also conducted by Akhavan et al. [13] for isotropic rectangular Mindlin plates under in-plane loadings and resting on Pasternak elastic foundation. Very Recently, Hosseini-Hashemi et al. [14] presented an analytical method for free vibration analysis of moderately thick FG rectangular plates supported by either Winkler or Pasternak elastic foundations. A new formula for the shear correction factors, used in the Mindlin plate theory, was obtained for FG plates. A new method of finite elements for analyzing Reissner– Mindlin FG plates based on the variational formulation was presented by Croce and Venini [15]. Zhao and Liew [16] were investigated the nonlinear response of functionally graded ceramic–metal plates under mechanical and thermal loads using the mesh-free kp-Ritz method. The nonlinear formulation is based on the first-order shear deformation plate theory (FSDT) and the von Karman strains, which deal with small strains and moderate rotations. Recently, Zhao et al. [17] have presented a free vibration analysis for FG square and skew plates with different boundary conditions using the element-free kp-Ritz method on the basis of the FSDT. Praveen and Reddy [18] analyzed the nonlinear transient response of FG plates that were subjected to a steady temperature field and lateral dynamic loads by using the first-order shear deformation plate theory and the finite element method. Although many studies on the vibration analysis of FG rectangular Mindlin plates have been carried out, the FSDT requires shear correction factor, depending on geometric parameters, boundary conditions, loading of the plate and the materials. This is due to the fact that the FSDT overestimates frequencies for thick FG plates, whereas third-order shear deformation plate theory (TSDT), higher-order shear deformation plate theory (HSDT) and the threedimensional (3-D) elasticity solution not only require no shear correction factor but also model a plate with smaller displacements and higher rigidity. Qian et al. [19,20] carried out an analysis of free and forced vibrations of both homogeneous and FG thick plates with the higher-order shear and normal deformable plate theory using meshless local Petrov–Galerkin method. Free vibration analysis of FG simply-supported square plates was studied by Pradyumna and Bandyopadhyay [21] using a higher order finite element formulation. Natural frequencies of FG square plates, with different boundary conditions at the edges, were obtained by Ferreira et al. [22] using the collocation method with multiquadric radial basis functions along with the FSDT and TSDT. Reddy [23] presented a theoretical formulation and finite element models for FG plates based on the TSDT having the thermo-mechanical coupling, time dependency, and von Karman-type geometric nonlinearity of the plates. The buckling and steady state vibrations of a simply supported functionally graded polygonal plate were carried out by Cheng and Batra [24] based on the TSDT. Zhong and Yu [25] employed a state-space approach to analyze free and forced vibrations of an FG piezoelectric rectangular thick plate simply supported at its edges. The free vibration of FG plates with different combinations of boundary conditions were studied by Roque et al. [26] who utilized the multiquadric radial basis function method and the HSDT. Matsunaga [27] presented natural frequencies and buckling stresses of simply supported FG plates based on 2D higher-order approximate plate theory. Using a semi-analytical approach, the exact 3-D solution for the thermo-elastic deformation of functionally graded simply-supported plates was

723

considered by Vel and Batra [28,29]. Furthermore, Vel and Batra [30] described an excellent investigation on the analytical solution for free and forced vibrations of FG simply-supported square plates based on the 3D elasticity solution. Kashtalyan [31] developed 3-D elasticity solution for a functionally graded simply-supported plate subjected to transverse loading. The effective material properties at a point were estimated by either the Mori–Tanaka [32,33] or the self-consistent schemes[34]. Unlike isotropic plates, the extension and bending equations for FG plates are coupled, resulting in five coupled differential equations in terms of five variables. As a result, to derive an exact closed-form solution becomes much more complicated. According to aforementioned survey of literature, it is found that most of research groups have made an endeavor to model FG rectangular plates on the basis of the CPT and solve them using numerical methods. To the best knowledge of the authors, no paper deals with exact closed-form solution for free vibration of FG rectangular thick plates based on the TSDT. The objective of this paper is to propose a new exact closedform approach for free vibration analysis of thick rectangular FG plates based on the third-order shear deformation theory of Reddy. The material properties are assumed to be graded through the thickness in accordance with a power-law distribution. Hamiltonian principle is used to extract the equations of dynamic equilibrium and natural boundary conditions of the plate. The exact closed-form solutions of the in-plane and out-of-plane displacements are obtained using the introduced potential functions and the separation variables method. To demonstrate the stability and accuracy of obtained results, the natural frequencies are compared with the available data in literature and 3-D finite element method (FEM) analysis. By comparing the results with the 3-D finite element model, it is obvious that the present exact solution can accurately and efficiently predict the in-plane and out-of-plane modes of FG plate. A new formula for the non-dimensional frequency parameter is proposed. In addition, the effects of the plate parameters such as thickness to length ratio, aspect ratio, the power law index and boundary conditions on the natural frequency parameters of FG rectangular plates are studied. 2. Mathematical formulation 2.1. Geometrical configuration A flat, and moderately thick functionally graded rectangular plate of length a, width b, and uniform thickness h, is depicted in Fig. 1. The plate has two opposite edges simply supported along

Fig. 1. Geometry of a rectangular plate composed of FGMs.

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x2 axis (i.e. along the edges x1 = 0 and x1 = a) while the other two edges may be free, simply supported or clamped. The Cartesian coordinate system (x1, x2, x3) is considered to obtain mathematical formulations when x1 and x2 axes are located in the mid-plane of the plate. 2.2. Material properties Functionally graded materials (FGMs) are composed of two kinds of materials: one is a metal and the other is ceramic. Here, it is assumed that the top surface of an FGM plate (x3 = h/2) is ceramic rich and bottom (x3 = h/2) is metal rich. Poisson’s ratio m is assumed to be constant and is taken as 0.3 throughout the analyses. The region between the two surfaces consists of material blended with both of them, which is assumed in the form of a simple power-law distribution as

Eðx3 Þ ¼ ðEc  Em ÞV f ðx3 Þ þ Em ;

ð1aÞ

qðx3 Þ ¼ ðqc  qm ÞV f ðx3 Þ þ qm ;

ð1bÞ

u3 ðx1 ; x2 ; x3 ; tÞ ¼ wðx1 ; x2 ; tÞ

ð3aÞ

ð3bÞ ð3cÞ

where u, v and w are the mid-plane displacements and w1, w2 denote the rotations of normal to mid-plane about the x2 and x1 axes, respectively and t is the time. It should be pointed out that traction-free boundary conditions are imposed on the top and bottom surfaces of the plate and the normal transverse stress r33 is assumed to be zero. The strain– displacement relations are given as

1 2

ð2Þ

where E(x3), q(x3) and Vf(x3) are Young’s modulus, mass density and the volume fraction of plate, respectively. The subscripts m and c represent the metallic and ceramic constituents, respectively. p is the power law index and takes only positive values. Typical values for metal and ceramics used in the FG plate are listed in Table 1. According to the comprehensive literature survey, most researchers use the power-law function as Eqs. (1) and (2) to describe the material properties and volume fraction of plate (e.g. see [14,17, 21,23,27]). Therefore, FGM plates with power-law function will be considered in this paper. It is noticeable that the rule of mixtures as stated by Eqs. (1) and (2), provides exact values for the mass density, q, and fairly good values of the young’s modulus E. Although a more accurate modeling of the macroscopic material properties of FGM requires a better understanding of the microstructure, the material properties calculated from Eqs. (1) and (2) may be applied for a good estimation of the macroscopic response of the functionally graded plate. In order to model the FGM more accurate, it is necessary to obtain the elastic stiffness using rigorous micromechanical estimates such as the Mori–Tanaka or self-consistent methods [32–34]. A similar analysis based on the micromechanical estimates will appear in a future communication. To study behavior of Eqs. (1a), (1b) and (2), the variation of Young’s modulus E in the thickness direction x3 for the Al/Al2O3 rectangular plate with various values of power law index p is shown in Fig. 2. For p = 0 and p = 1, the plate corresponds to the isotropic ceramic and metal, respectively; whereas the composition of metal and ceramic is linear for p = 1.

ð4Þ

Based on Hooke’s law, the stress–displacement relations are defined as

8 9 9 08 w1;1 þ mw2;2 > u;1 þ mv ;2 > > > > > > > > > > > > > > B> > > w2;2 þ mw1;1 > > v ;2 þ mu;1 > > > B> > > > < < = = B> 1m E B 1 m r12 ¼ 1m2 B 2 ðu;2 þ v ;1 Þ þ x3 2 ðw1;2 þ w2;1 Þ > > > > > B> > > > > > > > > > B> > 1m ðw2 þ w;2 Þ > > > > > > r23 > 0 > > > > > @> 2 > > > > > > > > > > > > : 1m : : ; ; ; ðw þ w Þ r13 0 ;1 1 2 8 9 0 > > > > > > > > = 2< 0 4x3  2 1m > 0 2 ðw2 þ w;2 Þ > h > > > > > > : 1m ; ðw þ w Þ ;1 1 2 8 91 ðw1;1 þ w;11 Þ þ mðw2;2 þ w;22 Þ > > > > > > > >C > > þ w Þ þ m ðw þ w Þ ðw > > C ;22 ;11 2;2 1;1 > > =C 3< 4x3 C 1 m  2 ðw1;2 þ w2;1 þ 2w;12 Þ C 2 > C 3h > > > > > > >C 0 > > A > > > > : ; 0 8 r11 9 > > > > > > > > > > r > > 22 > > < =

ð5Þ

where a comma followed by 1, 2 or 3 denotes the differentiation with respect to x1, x2 and x3 coordinate, E is the elasticity modulus and m is the Poisson’s ratio. 2.4. Equations of motion Herein, Hamilton’s principle is used to drive equations of motion based on the HSDT. The principle can be stated as follows:

2.3. The HSDT assumptions According to the HSDT, in which the in-plane displacements are expanded as cubic functions of the thickness coordinate and the transverse deflection is constant through the plate thickness, the displacement field is used as follows [37]:

Table 1 Material properties of the used FG plate.

Aluminum (Al) Alumina (Al2O3) Zirconia (ZrO2)

u2 ðx1 ; x2 ; x3 ; tÞ ¼ v ðx1 ; x2 ; tÞ þ x3 w2 ðx1 ; x2 ; tÞ   4x3 @wðx1 ; x2 ; tÞ  32 w2 ðx1 ; x2 ; tÞ þ @x2 3h

eij ¼ ðui;j þ uj;i Þ; i; j ¼ 1; 2; 3

 p x3 1 V f ðx3 Þ ¼ ; þ h 2

Material

u1 ðx1 ; x2 ; x3 ; tÞ ¼ uðx1 ; x2 ; tÞ þ x3 w1 ðx1 ; x2 ; tÞ   4x3 @wðx1 ; x2 ; tÞ  32 w1 ðx1 ; x2 ; tÞ þ @x1 3h

Z

t

Z

2h

0

3

E (GPa)

m

q (Kg/m )

70 380 200

0.3 0.3 0.3

2702 3800 5700

Z

b

Z

0

a



r11 de11 þ r22 de22 þ 2r12 de12 þ 2r13 de13

0

 þ 2r23 de23 Þdx1 dx2 dx3 dt d  2

Properties

þh2

Z 0

t

Z

þ2h

2h

Z 0

b

Z 0

a

!  2  2 2 qðzÞ u_ 1 þ u_ 2 þ u_ 3 dx1 dx2 dx3 dt ¼ 0 ð6Þ

Substituting Eqs. (3) and (4) into Eq. (6) and collecting the coefficients of du, dv, dw, dw1, dw2, the following equations of motion are obtained:

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Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

Fig. 2. Variation of Young’s modulus through the dimensionless thickness of Al/Al2O3 plate.

du : € @N1 @N6 € 1  4 I4 @ w € þ I22 w þ ¼ I1 u ð7aÞ 2 @x @x1 @x2 1 3h dv : € @N2 @N6 € 2  4 I4 @ w þ ¼ I1 v€ þ I22 w ð7bÞ 2 @x2 @x2 @x1 3h dw : !   @Q 1 @Q 2 4 @R1 @R2 4 @ 2 P1 @ 2 P6 @ 2 P2 þ 2 þ  þ þ2 þ @x1 @x2 h2 @x1 @x2 @x1 @x2 @x21 @x22 3h !  2   € € @2w € 4 @2w 4 @u @ v€ € I7 þ 2 þ 2 I4 þ ¼ I1 w 2 2 @x1 @x2 @x1 @x2 3h 3h ! € € 4 @ w1 @ w2 ð7cÞ þ 2 I55 þ @x1 @x2 3h dw1 :

  @M1 @M6 4 @P1 @P6 4 €1 € þ I33 w  Q 1 þ 2 R1 ¼ I22 u þ  2 þ @x1 @x2 3h @x1 @x2 h € 4 @w  2 I55 @x 1 3h dw2 :   @M2 @M6 4 @P2 @P6 4 €2  Q 2 þ 2 R2 ¼ I22 v€ þ I33 w þ  2 þ @x2 @x1 3h @x2 @x1 h € 4 @w  2 I55 @x 2 3h

ð7dÞ

I22 ¼ I2 

4 3h

I ;I 2 4 55

ðNi ; Mi ; P i Þ ¼ ðQ 2 ; R2 Þ ¼

Z

Z

þ2h

h2 þ2h

2h

r

Z

þ2h

2h

qð1; x3 ; x23 ; x33 ; x43 ; x63 Þdx3

¼ I5 

4 3h

I ;I 2 7 33

¼ I3 

8 3h

ri ð1; x3 ; x33 Þdx3 ; ði ¼ 1; 2; 6Þ

2 4 ð1; x3 Þdx3

I 2 5

ð8aÞ þ

16 9h

I 4 7

ð8bÞ

ð9aÞ ð9bÞ

þ2h

h2

r5 ð1; x23 Þdx3

ð9cÞ ð9dÞ

Based on the HSDT, the following boundary conditions are imposed for an edge parallel to, for example, x1-axis

N2 ¼ 0 or

v¼0

ð10aÞ

N6 ¼ 0 or u ¼ 0 4 M6  2 P6 ¼ 0 or w1 ¼ 0 3h 4 M2  2 P2 ¼ 0 or w2 ¼ 0 3h P2 ¼ 0 or w;2 ¼ 0   4 4 @P6 @P2 4 4 €  2 I4 v€  2 I5 w þ Q 2  2 R2 þ 2 2 2 @x @x 1 2 h 3h 3h 3h  2   € 4 @w € 2 ¼ 0 or w ¼ 0 þ I7 þw 2 @x 2 3h

ð10bÞ ð10cÞ ð10dÞ ð10eÞ

ð10fÞ

For generality and convenience, the following non-dimensional terms are introduced

x1 ; a

X2 ¼

x2 ; a

b a

h a

u a

v

g ¼ ; s ¼ ; u~ ¼ ; v~ ¼ ; a

I1 I2 I3 I1 ¼ ; I3 ¼ ; ; I2 ¼ qc a qc a2 qc a3 rffiffiffiffiffi 2 I4 a qc I5 I7 I4 ¼ ; b¼x ; I7 ¼ ; ; I5 ¼ qc a4 h Ec qc a5 qc a7 I22 I33 I55 I22 ¼ ; I33 ¼ ; I55 ¼ qc a2 qc a3 qc a5 ~1 ¼ w ; w 1

ð7eÞ

Z

r11 ¼ r1 ; r22 ¼ r2 ; r23 ¼ r4 ; r13 ¼ r5 ; r12 ¼ r6

X1 ¼

where dot-overscript convention indicates the differentiation with respect to the time variable t. The stress resultants Ni, Mi, Pi, Qi, Rsi and the inertias Ii (i = 1, 2, 3, 4, 5, 7) are defined by

ðI1 ; I2 ; I3 ; I4 ; I5 ; I7 Þ ¼

ðQ 1 ; R1 Þ ¼

e ¼ w

w ; a

~2 ¼ w ; w 2

ð11Þ

where x is the natural frequency of the plate. Substituting Eq. (5) into Eqs. (9a)–(9c), the non-dimensional stress resultants are expressed as

e 1 ¼ Aðu ~ 2;2 Þ  4 Gðw ~ 1;1 þ mw ~ 1;1 þ w ~ ;1 þ mv~ ;2 Þ þ Bðw e ;11 N 3s2 ~ 2;2 þ w e ;22 ÞÞ þ mðw

ð12aÞ

e 2 ¼ Aðmu ~ 1;1 þ w ~ 2;2 Þ  4 Gðmðw ~ 1;1 þ w e ;11 Þ ~ ;1 þ v~ ;2 Þ þ Bðmw N 3s2 ~ 2;2 þ w e ;22 Þ þw

ð12bÞ

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  e 6 ¼ 1  m Aðu ~ 1;2 þ w ~ 2;1 Þ  4 Gðw ~ 1;2 þ w ~ 2;1 þ 2 w ~ ;2 þ v~ ;1 Þ þ Bðw e ;12 Þ N 3s2 2 ð12cÞ e 1 ¼ Bðu ~ 2;2 Þ  4 Fðw ~ 1;1 þ mw ~ 1;1 þ w ~ ;1 þ mv~ ;2 Þ þ Dðw e ;11 M 3s2 ~ 2;2 þ w e ;22 ÞÞ þ mðw

ð12dÞ

e 2 ¼ Bðmu ~ 1;1 þ w ~ 2;2 Þ  4 Fðmðw ~ 1;1 þ w e ;11 Þ ~;1 þ v~ ;2 Þ þ Dðmw M 3s2 ~ 2;2 þ w e ;22 Þ þw

ð12eÞ

  e 6 ¼ 1  m Bðu ~ 1;2 þ w ~ 2;1 Þ  4 Fðw ~ 1;2 þ w ~ 2;1 þ 2 w ~;2 þ v~ ;1 Þ þ Dðw e ;12 Þ M 3s2 2 ð12fÞ e 1 ¼ Gðu ~ 2;2 Þ  4 Hðw ~ 1;1 þ mw ~ 1;1 þ w ~ ;1 þ mv~ ;2 Þ þ Fðw e ;11 P 3s2 ~ 2;2 þ w e ;22 ÞÞ þ m ðw e 2 ¼ Gðmu ~ 1;1 þ w ~ 2;2 Þ  4 Hðmðw ~ 1;1 þ w ~ ;1 þ v~ ;2 Þ þ Fðmw e ;11 Þ P 3s2 ~ 2;2 þ w e ;22 Þ þw

ð12gÞ

ð12hÞ

  e 6 ¼ 1  m Gðu ~ 1;2 þ w ~ 2;1 Þ  4 Hðw ~ 1;2 þ w ~ 2;1 þ 2 w ~;2 þ v~ ;1 Þ þ Fðw e ;12 Þ P 3s2 2 ð12iÞ   e 1 ¼ 1  m A  4 D ðw ~1 þ w e ;1 Þ Q 2 2 s

ð12jÞ

   4 D e2 ¼ 1  m A  ðw ~2 þ w e ;2 Þ Q 2 2 s

ð12kÞ

ð12lÞ

  e 2 ¼ 1  m D  4 F ðw ~2 þ w e ;2 Þ R 2 s2

ð12mÞ

Z

h=2

h=2

A¼

A ; Ec a

F¼

F ; Ec a5

B¼

e 1 ¼ N1 ; N Ec a e 1 ¼ M1 ; M Ec a2

B ; Ec a 2

H¼

D ; Ec a3

G¼

G ; Ec a4

H Ec a7

e 2 ¼ N2 ; N Ec a

ð13bÞ e 6 ¼ N6 N Ec a

e 2 ¼ M2 ; M Ec a2

e 1 ¼ P1 ; P Ec a4

e 2 ¼ P2 ; P Ec a4

e 1 ¼ Q1 ; Q Ec a

e 2 ¼ Q2 ; Q Ec a

ð13aÞ

ð13cÞ

e 6 ¼ M6 M Ec a2

ð13dÞ

e 6 ¼ P6 P Ec a4 e 1 ¼ R1 ; R Ec a3

ð13eÞ e 2 ¼ R2 R Ec a3

ð13fÞ

For free vibration, we assume that the three coordinates w1, w2 and w vary harmonically with respect to the time variable t as follows:

~ 2 ðX 1 ; X 2 ; tÞ ¼ w ðx1 ; x2 Þeixt w 2 ð14Þ

wðx1 ; x2 Þ ixt e a

pffiffiffiffiffiffiffi where i ¼ 1. Since the present paper deals with the vibration of FG rectangular plates, the in-plane forces Ni (i = 1, 2, 6), moments Mi (i = 1, 2, 6) and higher order stress resultants Pi (i = 1, 2, 6) are coupled due to existence of B and G terms. Then, Eqs. (7a)–(7e) should be simultaneously solved. As a result, by substituting Eqs. (12a)–(12m) into Eqs. (7a)–(7e), the equations of motion are given in dimensionless form as follows:

  1m 2 1þm ~ ;1 þ v~ ;2 Þ;1 A r u~ þ ðu 2 2    4 1  m 2~ 1þm ~ ~ 2;2 Þ þ B 2G r w1 þ ðw1;1 þ w ;1 3s 2 2 4  ;1  2 Gr2 w 3s ~ 1 þ 4 I4 b2 w ~  I22 s2 b2 w  ;1 ¼ I1 s2 b2 u 3     1m 2 1þm 4 ~ ;1 þ v~ ;2 Þ;2 þ B  A r v~ þ G ðu 3s2 2 2   1  m 2~ 1þm ~ 4 ~ 2;2 Þ  ;2  2 Gr2 w r w2 þ ðw1;1 þ w  ;2 3s 2 2 ~ 2 þ 4 I4 b2 w  ;2 ¼ I1 s2 b2 v~  I22 s2 b2 w 3 ! ! 4F 16H 2 ~ ~ 2;2 Þ þ 1  m A þ 16F  8D  r ð w þ w 1;1 3s2 9s4 2 s4 s2

eþ ¼ I1 s2 b2 w

Eðx3 Þ ð1; x3 ; x23 ; x33 ; x43 ; x63 Þdx3 1  m2 D¼

e 1 ; X 2 ; tÞ ¼ wðX

~ 1;1 þ w ~ 2;2 þ r2 wÞ e   ðw

  e 1 ¼ 1  m D  4 F ðw ~1 þ w e ;1 Þ R 2 s2

ðA; B; D; G; F; HÞ ¼

~ 1 ðX 1 ; X 2 ; tÞ ¼ w ðx1 ; x2 Þeixt ; w 1

ð15aÞ

ð15bÞ

16H 4 4G 2 eþ r w r ðu~ ;1 þ v~ ;2 Þ 9s4 3s 2

16 4 ~ ;1 þ v~ ;2 Þ e  I4 b2 ðu I7 b2 r2 w 9s2 3

4 ~ 1;1 þ w ~ 2;2 Þ  I55 b2 ðw 3 ! 16H 4  ;1  F r2 w 9s4 3s2 !  8F 16H 1  m 2~ 1þm ~ ~ 2;2 Þ ðw1;1 þ w þ D 2 þ 4 r w1 þ ;1 3s 9s 2 2 !  4G 1m 2 1þm ~ ;1 þ v~ ;2 Þ;1 r u~ þ þ B 2 ðu 3s 2 2 ! 1  m 8D 16F ~ e ;1 Þ  A  4 ðw þ 1 þ w 2 s2 s ~ 1 þ 4 I55 b2 w ~  I33 s2 b2 w  ;1 ¼ I22 s2 b2 u 3 ! 16H 4  ;2  F r2 w 9s4 3s2 !  8F 16H 1  m 2~ 1þm ~ ~ w þ D 2þ r þ þ w Þ ð w 2 1;1 2;2 ;2 3s 9s4 2 2 !  4G 1m 2 1þm ~ ;1 þ v~ ;2 Þ;2 c þ B 2 r v~ þ ðu 3s 2 2

ð15cÞ

ð15dÞ

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Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

!

þ

/2 between two obtained equations. After simplifying and collecting terms, we reach the following equation:

1  m 8D 16F ~ e ;2 Þ  A  4 ðw 2 þ w 2 s2 s

~ 2 þ 4 I55 b2 w  ;2 ¼ I22 s2 b2 v~  I33 s2 b2 w 3

ð15eÞ

where r2 denotes the Laplace operator, given by

@2

2

r ¼

@X 21

þ

@2

4

@X 22

;r ¼

@4 @X 41

þ2

@4 @X 21 @X 22

þ

@4 @X 42

e þ a2 r6 w e þ a3 r4 w e þ a4 r2 w e þ a5 w e ¼0 a1 r8 w

ð22Þ

where ai (i = 1–5) are constants defined in Appendix A. Eq. (22) can be solved as

e ¼ W1 þ W2 þ W3 þ W4 w ð16Þ

ð23Þ

where the potential functions W1, W2, W3 and W4 are defined as

r2 W 1 þ a21 W 1 ¼ 0

ð24aÞ

r2 W 2 þ a22 W 2 ¼ 0

ð24bÞ

Herein, we introduce the functions f1, f2, f3, f4 and /1, /2 as follows:

r2 W 3 þ a23 W 3 ¼ 0

ð24cÞ

  1m 4 1m~ ~þ B f1 ¼ A G u w1 3s2 2 2

ð17aÞ

r2 W 4 þ a24 W 4 ¼ 0

ð24dÞ

  1m 4 1m~ v~ þ B  2 G w2 3s 2 2

ð17bÞ

2.5. Exact solution procedure

f2 ¼ A

!

!

f3 ¼

B

4G 1  m 8F 16H 1  m ~ ~þ D þ u w1 3s2 3s2 9s4 2 2

f4 ¼

B

! ! 4G 1  m 8F 16H 1  m ~ ~ þ D v þ w2 3s2 3s2 9s4 2 2

and a21 ; a22 ; a23 ; a24 are the roots of the following fourth-order equation:

a1 z4 þ a2 z3 þ a3 z2 þ a4 z þ a5 ¼ 0 Then, it can be written as follows:

ð17cÞ

a21 ¼

ð17dÞ ð18aÞ

/2 ¼ f3;1 þ f4;2

ð18bÞ

a22 ¼

By using Eqs. (17a)–(17d), (18b) in Eqs. (15a)–(15e), the equation of motions can be rewritten as follows:

r2 f 1 þ

1þm e ;1 þ y4 r2 w e ;1 / ¼ y1 f1 þ y2 f3 þ y3 w 1  m 1;1

ð19aÞ

r2 f 2 þ

1þm e ;2 þ y4 r2 w e ;2 / ¼ y1 f2 þ y2 f4 þ y3 w 1  m 1;2

ð19bÞ

r2 f 3 þ

1þm e ;1 þ y8 r2 w e ;1 / ¼ y5 f1 þ y6 f3 þ y7 w 1  m 2;1

ð19cÞ

r2 f 4 þ

1þm e ;2 þ y8 r2 w e ;2 / ¼ y5 f2 þ y6 f4 þ y7 w 1  m 2;2

ð19dÞ

y9 r /1 þ y10 r /2 þ y11 /1 þ y12 /2 ð19eÞ

where yi(i = 1  15)are constants governed by material properties, non-dimensional frequency b and structural geometry, given in Appendix A. After differentiating Eqs. (19a) and (19b) with respect to X1 and X2, respectively, two obtained equations should be added together. Thus, we have

2 e þ y 4 r4 w e r2 /1 ¼ y1 /1 þ y2 /2 þ y3 r2 w 1m

ð20Þ

Similarly, using of Eqs. (19c) and (19d) leads to

2 e þ y 8 r4 w e r2 /2 ¼ y5 /1 þ y6 /2 þ y7 r2 w 1m

a2 1 1  C2 þ 4a1 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  3  a22 4a3 21=3 ða23  3a2 a4 þ 12a1 a5 Þ a C1 4a2 a3 8a4    1=3   23 þ 2  ð4C 2 Þ  2 a1 3a1 C 1 2a1 3a1 a1 a1 32 a1

ð26bÞ a23 ¼

a2 1 1 þ C2  2 4a1 2 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3  a22 4a3 21=3 ða23  3a2 a4 þ 12a1 a5 Þ a2 4a2 a3 8a4 C1     þ  þ  Þ ð4C 2 3a1 C 1 a1 2a21 3a1 a31 a21 321=3 a1

ð26cÞ a24 ¼

a2 1 1 þ C2 þ 4a1 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   3  1=3  2 2 a3  3a2 a4 þ 12a1 a5 a2 4a3 2 C1 a 4a2 a3 8a4     1=3 þ  23 þ 2  ð4C 2 Þ 2 a1 3a1 C 1 2a1 3a1 a1 a1 32 a1

where i ¼

2

e þ y14 r2 w e þ y15 w e ¼ y13 r4 w

a2 1 1  C2  2 4a1 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   3  1=3 a3  3a2 a4 þ 12a1 a5 a22 4a3 2 a 4a2 a3 8a4 C1     1=3   23 þ 2  ð4C 2 Þ 2 a1 3a1 C 1 2a1 3a1 a1 a1 32 a1

ð26aÞ

/1 ¼ f1;1 þ f2;2

2

ð25Þ

ð21Þ

pffiffiffiffiffiffiffi 1 and

ð26dÞ

   C 1 ¼ 2a33  9a2 a3 a4 þ 27 a1 a24 þ a5 a22  72a1 a3 a5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13  3  2 þ 4 a23  3a2 a4 þ 12a1 a5 þ 2a33  9a2 a3 a4 þ 27ða1 a24 þ a5 a22 Þ  72a1 a3 a5

ð27aÞ ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1=3  2 a3  3a2 a4 þ 12a1 a5 a22 2a3 2 C1 C2 ¼  þ þ 1=3 3a1 C 1 4a21 3a1 32 a1

ð27bÞ

Using of Eqs. (19e), (21) and (22), the introduced functions /1 and /2 can be given by

e þ e2 r4 w e þ e 3 r2 w e þ e4 w e /1 ¼ e1 r6 w

ð28aÞ

e þ e6 r4 w e þ e 7 r2 w e þ e8 w e /2 ¼ e5 r6 w

ð28bÞ

Substituting Eqs. (24a)–(24d) into Eqs. (28a) and (28b) yields e 2.5.1. Obtaining of w e In order to solve Eqs. (15a)–(15e), it is necessary to obtain w, firstly. To this end, we introduce /1 from Eq. (19e) into Eqs. (20) e is obtained by Eliminating and (21). The transverse deflection w

/1 ¼ d1 W 1 þ d2 W 2 þ d3 W 3 þ d4 W 4

ð29aÞ

/2 ¼ d5 W 1 þ d6 W 2 þ d7 W 3 þ d8 W 4

ð29bÞ

where di (i = 1–8) are listed in Appendix A.

728

Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

~ 1 and w ~2 ~; v ~; w 2.5.2. Obtaining of u Substituting Eqs. (24a)–(24d) and (29a) and (29b) into Eqs. (19a)–(19e), the alternative functions f1, f2, f3 and f4 may be then expressed as

~2 ¼ w

ð24Gs2  18Bs4 Þf2 þ 18As4 f4  ð1 þ mÞ 16G2  16AH þ 24AF s2  24BGs2 þ 9B2 s4  9ADs4 ð37dÞ

f1 ¼ C 5 W 1;1 þ C 6 W 2;1 þ C 7 W 3;1 þ C 8 W 4;1 þ C 9 W 5;2 þ C 10 W 6;2 ð30aÞ f2 ¼ C 5 W 1;2 þ C 6 W 2;2 þ C 7 W 3;2 þ C 8 W 4;2  C 9 W 5;1  C 10 W 6;1 ð30bÞ f3 ¼ C 1 W 1;1 þ C 2 W 2;1 þ C 3 W 3;1 þ C 4 W 4;1 þ W 5;2 þ W 6;2

ð30cÞ

f4 ¼ C 1 W 1;2 þ C 2 W 2;2 þ C 3 W 3;2 þ C 4 W 4;2  W 5;1  W 6;1

ð30dÞ

where the potential functions W5 and W6 are defined as

r2 W 5 þ a25 W 5 ¼ 0

ð31aÞ

r2 W 6 þ a26 W 6 ¼ 0

ð31bÞ

    W 1 ¼ A1 Sinh l1 X 2 þ A2 Cosh l1 X 2 Sinðn1 X 1 Þ

    þ B1 Sinh l1 X 2 þ B2 Cosh l1 X 2 Cosðn1 X 1 Þ W 2 ¼ ½A3 Sinhðl2 X 2 Þ þ A4 Coshðl2 X 2 ÞSinðn2 X 1 Þ þ ½B3 Sinhðl2 X 2 Þ þ B4 Coshðl2 X 2 ÞCosðn2 X 1 Þ

þ ½B7 Sinhðl4 X 2 Þ þ B8 Coshðl4 X 2 ÞCosðn4 X 1 Þ ð33bÞ

W 5 ¼ ½A9 Sinhðl5 X 2 Þ þ A10 Coshðl5 X 2 ÞCosðn5 X 1 Þ þ ½B9 Sinhðl5 X 2 Þ þ B10 Coshðl5 X 2 ÞSinðn5 X 1 Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 þ c21  4c2

ð34aÞ

q2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1  c21  4c2

ð34bÞ

2

The coefficients Ci (i = 1–10) are given as follows:

8 ciþ2 > ; i ¼ 1—4 > 4 2 > > a  c 1 ai þ c2 > i > > < y  y a2 þ C y þ a2 ð1 þ mÞ þ d ð1 þ mÞ i4 i 7 8 i4 6 i4 ; i ¼ 5—8 Ci ¼  > y5 ð1 þ mÞ > > > > > y þ a2i4 > : 6 ; i ¼ 9; 10 y5 ð35Þ where

 1  y3 y5  y1 y7  ðy4 y5 þ y7  y1 y8 Þa2i2 þ y8 a4i2 ð1 þ mÞ 1 þ m    þ di2 y5 ð1 þ mÞ  diþ2 y1 þ a2i2 ð1 þ mÞ ; i ¼ 3; 4; 5; 6 ð36Þ

ð38eÞ

W 6 ¼ ½A11 Sinhðl6 X 2 Þ þ A12 Coshðl6 X 2 ÞCosðn6 X 1 Þ þ ½B11 Sinhðl6 X 2 Þ þ B12 Coshðl6 X 2 ÞSinðn6 X 1 Þ

ð38fÞ

where

a21 ¼ l21 þ n21 ; n21 > 0; l21 < 0

ð39aÞ

a22 ¼ l22 þ n22 ; n22 > 0; l22 < 0

ð39bÞ

a23 ¼ l23 þ n23 ; n23 > 0; l23 > 0

ð39cÞ

2 4

2 4

n24 ;

n24

2 5

2 5

n25 ;

n25

2 6

2 6

n26 ;

n26

a ¼l þ a ¼l þ a ¼l þ

2 4

ð39dÞ

2 5

ð39eÞ

l <0

> 0;

l <0 > 0; l26 < 0 > 0;

ð39fÞ

2.5.4. The classical boundary conditions Based on the TSDT, the classical boundary conditions may be obtained for an edge parallel to, for example, X1-axis as the following non-dimensional equations: – Simply support

Finally, the exact closed-form displacement field of the plate according to Reddy’s theory, is obtained by substituting Eqs. (30a)–(30d) into Eqs. (17a)–(17c) and (17d)

~ ¼ 0; u

  32H  48F s2 þ 18Ds4 f1 þ ð24Gs2  18Bs4 Þf3  ð1 þ mÞ 16G2  16AH þ 24AF s2  24BGs2 þ 9B2 s4  9ADs4

– Clamped

~¼ u

ð38dÞ

ð33aÞ

Clearly by solving Eq. (32)

ci ¼

ð38cÞ

W 4 ¼ ½A7 Sinhðl4 X 2 Þ þ A8 Coshðl4 X 2 ÞSinðn4 X 1 Þ

c1 ¼ y1  y6 c2 ¼ y2 y5 þ y1 y6

a25 ¼

ð38bÞ

þ B6 Cosðl3 X 2 ÞCosðn3 X 1 Þ

ð32Þ

where

a24 ¼

ð38aÞ

W 3 ¼ ½A5 Sinðl3 X 2 Þ þ A6 Cosðl3 X 2 ÞSinðn3 X 1 Þ þ ½B5 Sinðl3 X 2 Þ

and a25 ; a26 are the roots of the following quadratic equation:

r 2 þ c1 r þ c2 ¼ 0

2.5.3. The separation variables method By virtue of the separation variables method, one set of solutions for Eqs. (24a)–(24d) and Eqs. (31a) and (31b) can be written as

e 2 ¼ 0; N

e ¼ 0; w

~ 1 ¼ 0; w

e 2 ¼ 0; M

e2 ¼ 0 P ð40aÞ

~ ¼ 0; u

v~ ¼ 0;

e ¼ 0; w

~ 1 ¼ 0; w

~ 2 ¼ 0; w

e ;2 ¼ 0 w ð40bÞ

ð37aÞ – Free

~1 ¼ w

ð24Gs2  18Bs4 Þf1 þ 18As4 f3

 ð1 þ mÞ 16G2  16AH þ 24AF s2  24BGs2 þ 9B2 s4  9ADs4

e 6 ¼ 0; N

e 2 ¼ 0; N

e 2 ¼ 0; M

ð37bÞ

v~ ¼

ð32H  48F s þ 18Ds Þf2 þ ð24Gs  18Bs Þf4  ð1 þ mÞ 16G2  16AH þ 24AF s2  24BGs2 þ 9B2 s4  9ADs4 2

4

2

4

ð37cÞ

e e e2  4 R e þ 4 2 @ P6 þ @P2 Q s2 2 3s2 @X 1 @X 2   e 16 @w ~2 ¼ 0  2 b2 I7 þw 9s @X 2

e 2 ¼ 0; P

!

e6 ¼ 0 e6 4 P M 3s2 ð40cÞ

4 4 ~2 þ I4 b2 v~ þ b2 I5 w 3 3

729

Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

By changing subscripts 1 and 2 in Eqs. (40a)–(40c), the different boundary conditions are obtained for the edges X1 = 0 and X1 = 1.

Table 2 pffiffiffiffiffiffiffiffiffiffiffiffi Comparison of fundamental frequency parameter b ¼ xa2 qc =Ec =h for SSSS Al/Al2O3 square plates (g = 1).

2.6. Lévy-type solution

Material

Method

In this Section we will develop the Lévy-type solutions corresponding to the cases where two opposite edges are simply supported and the remaining edges of the plate can have any boundary conditions. According to Fig. 1, the boundary conditions of plate at X1 = 0 and X1 = 1 are simply supported, then Eqs. (38a)–(38f) may be written as

Fully ceramic

Present

W 1 ¼ ½A1 Sinhðl1 X 2 Þ þ A2 Coshðl1 X 2 ÞSinðn1 X 1 Þ

ð41aÞ

W 2 ¼ ½A3 Sinhðl2 X 2 Þ þ A4 Coshðl2 X 2 ÞSinðn2 X 1 Þ

ð41bÞ

W 3 ¼ ½A5 Sinðl3 X 2 Þ þ A6 Cosðl3 X 2 ÞSinðn3 X 1 Þ

ð41cÞ

W 4 ¼ ½A7 Sinhðl4 X 2 Þ þ A8 Coshðl4 X 2 ÞSinðn4 X 1 Þ

ð41dÞ

W 5 ¼ ½A9 Sinhðl5 X 2 Þ þ A10 Coshðl5 X 2 ÞCosðn5 X 1 Þ

ð41eÞ

W 6 ¼ ½A11 Sinhðl6 X 2 Þ þ A12 Coshðl6 X 2 ÞCosðn6 X 1 Þ

ð41fÞ

where

n1 ¼ n2 ¼ n3 ¼ n4 ¼ n5 ¼ n6 ¼ mp;

m ¼ 1; 2; 3; . . .

ð42Þ

Substituting Eqs. (30a)–(30d) into three appropriate boundary conditions (i.e., Eqs. (40a)–(40c)) along the edges X2 = 0 and X2 = g leads to a coefficient matrix. For a nontrivial solution, the determinant of the coefficient matrix must be set to zero for each m. Solving the eigenvalue equations yields the frequency parameters b.

3. Numerical results Using the previously developed analytical exact solutions based on the third-order shear deformation plate theory, numerical parametric studies are carried out. Two types of FG plates are used in this study which their material properties are listed in Table 1. Chi and Chung [35] showed that Poisson’s ratio affects on the mechanical behavior of the FG plates slightly, therefore it can be assumed to be constant. In all FSDT comparison results, the shear correction factor has been taken to be 5/6. Identification of this factor in FGMs is investigated by Nguyen et al. [36]. It should be noted that notation SCSF, for example, denotes that the edges x1 = 0, x2 = 0, x1 = a, and x2 = b are simply supported, clamped and free, respectively. Also, a well-known commercially available FEM package was used for the extraction of the frequency parameters. 3.1. Comparison results In this section, according to a beneficial literature review, two comparison studies are provided to validate the results of the present study and demonstrate its accuracy. 3.1.1. Isotropic square plate In order to investigate the efficiency and stability of the present exact solution, the results are compared with those obtained for isotropic plate as shown in Table 2. According to Eqs. (1) and (2), when the power law index p, approaches zero or infinity, the plate is isotropic composed of fully ceramic or metal, respectively. Three fundamental frequency parameters b of SSSS Al/Al2O3 square plates (g = 1) are presented in Table 2 for s = 0.1 and 0.2. The results are compared with those obtained by Shufrin and Eisenberger [38] based on the HSDT. It is found that when gradient index approaches zero or infinity, the frequency parameters of FG plate converge to relevant isotropic one. Excellent agreement among the results confirms the high accuracy of the current analytical approach.

s = h/a 0.1

0.2

p = 103 p = 104 p = 105 p = 106

5.7672 5.7692 5.7694 5.7694 5.7694

5.2794 5.2811 5.2813 5.2813 5.2813

p = 102 p = 103 p = 104 p = 105

3.09729 2.9556 2.9394 2.9376 2.9376

2.8215 2.7042 2.6907 2.6891 2.6891

HSDT [38] Fully metallic

Present

HSDT [38]

3.1.2. FGM square plate Table 3 shows a comparison of the frequency parameters pffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ xh q =Ec for SSSS Al/Al2O3 square thin and moderately thick b c plates (s = 0.001, 0.1 and 0.2) with those obtained by Hosseini-Hashemi et al. [14], Zhao et al. [17] and Matsunaga [27] when p = 0, 0.5, 1, 4, 10 and 1. In addition, the corresponding mode shapes m and n, denoting the number of half-waves in the x1- and x2direction, respectively, are presented for any of the frequency ^ Also, in Table 4, the exact solution procedure is valparameters b. idated by comparing the ffievaluation of fundamental frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ xh q =Em for a simply supported Al/ZrO2 square parameters b m plates with those of two-dimensional higher-order theory [27], three-dimensional theory by employing the power series method [30], finite element HSDT method [21], finite element FSDT method [21] and an analytical FSDT solution [14]. From Tables 3 and 4, It is evident that there is a very good agreement among the results confirming the high accuracy of the current analytical approach. As it is seen, the present exact solution is reported the good agreement with those obtained bypthe  ffiffiffiffiffiffi HSDT [27] for the thicker FG square plates s ¼ 0:1; 0:2; 1= 10 particularly at the higher modes of vibration. The difference between the present exact natural frequencies with those obtained by the 3-D method [30] may be due to the estimation of material properties of FG plates. In Ref. [30], the functionally graded material properties at a point were expressed by the local volume fractions and the material properties of the phases using two methods: Mori–Tanaka [32,33] and the self-consistent scheme [34], whereas, in the present exact analysis, The properties of the plate are assumed to vary through the thickness of the plate with a power-law distribution of the volume fractions of the two materials between the two surfaces. The difference between the present exact solution with those obtained by the analytical FSDT solutions [14] caused by vanishing of in-plane displacement components of FG plate in Ref. [14]. In fact, as the present exact procedure provided, in-plane displacement components u and v should be taken into account and are coupled with transverse displacement components w, w1 and w2. 3.2. Results and discussion This section describes the free vibration analysis of FG rectangular plates to validate approximate two-dimensional theories and new computational techniques in future. Unless mentioned otherwise, all natural frequencies of FG rectangular plates are pffiffiffiffiffiffiffiffiffiffiffiffi considered to be dimensionless as b ¼ xa2 =h qc =Ec (called the qffiffiffiffiffiffiffiffiffi  ¼ xa2 =h q  =E (called the eigenfrefrequency parameter) and as b  are Young’s modulus and quency parameter) in which E and q mass density of the plate in the mid-plane (x3 = 0), respectively. It should be noted that the eigenfrequency parameter

730

Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

Table 3 pffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ xh q =Ec for SSSS Al/Al2O3 square plates (g = 1). Comparison of the natural frequency parameter b c

s = h/a

(m, n)a

Method

0

0.5

1

4

10

1

0.05

(1, 1)

Present FSDT [14] FSDT [17]

0.0148 0.0148 0.0146

0.0125 0.0128 0.0124

0.0113 0.0115 0.0112

0.0098 0.0101 0.0097

0.0094 0.0096 0.0093

– – –

0.1

(1, 1)

Present HSDT [27] FSDT [14] FSDT [17] Present HSDT [27] FSDT [17] Present HSDT [27] FSDT [17]

0.0577 0.0577 0.0577 0.0568 0.1377 0.1381 0.1354 0.2113 0.2121 0.2063

0.0490 0.0492 0.0492 0.0482 0.1174 0.1180 0.1154 0.1807 0.1819 0.1764

0.0442 0.0443 0.0445 0.0435 0.1059 0.1063 0.1042 0.1631 0.1640 0.1594

0.0381 0.0381 0.0383 0.0376 0.0903 0.0904 – 0.1378 0.1383 –

0.0364 0.0364 0.0363 0.3592 0.0856 0.0859 0.0850 0.1301 0.1306 0.1289

0.0293 0.0293 0.0294 – 0.0701 0.0701 – 0.1076 0.1077 –

Present HSDT [27] FSDT [14] FSDT [17] Present HSDT [27] Present HSDT [27]

0.2113 0.2121 0.2112 0.2055 0.4623 0.4658 0.6688 0.6753

0.1807 0.1819 0.1806 0.1757 0.3989 0.4040 0.5803 0.5891

0.1631 0.1640 0.1650 0.1587 0.3607 0.3644 0.5254 0.5444

0.1378 0.1383 0.1371 0.1356 0.2980 0.3000 0.4284 0.4362

0.1301 0.1306 0.1304 0.1284 0.2771 0.2790 0.3948 0.3981

0.1076 0.1077 0.1075 – 0.2355 0.2365 0.3407 0.3429

(1, 2)

(2, 2)

0.2

(1, 1)

(1, 2) (2, 2) a

Power law index (p)

m and n are wave numbers in direction x1 and x2, respectively.

Table 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ xh q =Em for SSSS Al/ZrO2 square plates (g = 1). Comparison of fundamental frequency parameter b m Method

p=0

Present HSDT [27] 3-D [30] HSDT [21] FSDT [21] FSDT [14]

s = 0.2

p=1 pffiffiffiffiffiffi

s ¼ 1= 10

s = 0.1

s = 0.05

s = 0.1

s = 0.2

p=2

p=3

p=5

0.4623 0.4658 0.4658 0.4658 0.4619 0.4618

0.0577 0.0577 0.0577 0.0578 0.0577 0.0576

0.0158 0.0158 0.0153 0.0157 0.0162 0.0158

0.0619 0.0619 0.0596 0.0613 0.0633 0.0611

0.2276 0.2285 0.2192 0.2257 0.2323 0.2270

0.2256 0.2264 0.2197 0.2237 0.2325 0.2249

0.2263 0.2270 0.2211 0.2243 0.2334 0.2254

0.2272 0.2281 0.2225 0.2253 0.2334 0.2265

qffiffiffiffiffiffiffiffiffi  ¼ xa2 =h q  =E is defined for the first time and has its special own b characteristics, as will be shown in Sections 3.2.3 and 3.2.4. 3.2.1. The in-plane and out-of-plane modes The vibration modes of FG plate may be divided into two main categories: the out-of-plane (transverse) modes and the in-plane modes. For the in-plane modes, the magnitude of transverse displacement is very smaller than in-plane displacements, u and v. There is a main difference between in-plane modes of isotropic plates and FG ones. When an isotropic plate has in-plane mode, there is no transverse displacement and plate can only move along in-plane directions but because of existing coupling between inplane and out of plane vibration in FG plate, in-plane mode includes two kinds of motion whereas in-plane vibration is dominant. In Tables 5–7, based on the present exact closed-form solutions and finite element method, numerical results have been performed for SSSS, SCSC and SFSF Al/Al2O3 square plates (g = 1) when p = 1. A well-known commercially available FEM package was used to investigate 3D free vibration of FG square plates. A convergence study was first conducted to ensure independency of FEM results from the number of elements and then their high accuracy was examined through solving some problems of the literature. The length of square plates is 1 m. Three different thicknesses 0.05 m (corresponding to thin plates), 0.1 m and 0.2 m (corresponding to moderately thick plates) and 0.3 m (corresponding to thick plates) have been used. All of calculations are obtained for first ten natural frequencies. Bold-faced values in Tables 5–7

indicate the in-plane modes. The percentage difference given in Tables 5–7 is defined as follows:

%Diff ¼

½FEM  ðExactHSDTÞ  100 FEM

An excellent agreement is observed between the present exact HSDT and the FEM. It is seen that the exact results of proposed method are close to the FE analysis results at lower and higher frequencies. Frequencies rise with an increase in the thickness of plate. This phenomena originates from the increasing the rigidity of plate. Frequencies decrease when less restraining boundary is used at the edges of square plates. This is due to the fact that higher constraints at the edges increase the flexural rigidity of the plate, leading to a higher frequency response. From the results of Tables 5–7, it is observed that the present solution can predict the in-plane modes, excellently. Also, the results reveal that the number of the existing in-plane modes in the first ten natural frequencies, is considerably increased by increasing the thickness of plate. This feature is due to the fact that the bending energy of plate has significant sensibility respect to the thickness. It is to be reminded that the out-ofplane modes depend on the bending energy, directly. Then, the number of the out-of-plane modes is reduced by increasing the plate thickness and the number of in-plane modes is increased. 3.2.2. Identification of coupling effects on the frequency parameters b Eqs. (15a)–(15e) prove the fact that the coupling between the in-plane and out-of-plane components is due to the potential coupling (because of the existence of B and G terms) and the kinetic

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Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735 Table 5 First 10 natural frequency (Hz) for SSSS Al/Al2O3 square plates (g = 1, p = 1).

s ¼ ha

Method

0.05

Present FEM Error (%) Present FEM Error (%) Present FEM Error (%) Present FEM Error (%)

0.1

0.2

0.3

Mode 1

2

3

4

5

6

7

8

9

10

359.97 357.37 0.728 703.45 699.30 0.593 1298.3 1296.3 0.154 1753.6 1759.8 +0.352

889.39 883.58 0.658 1685.7 1679.7 0.357 2575.5 2574.6 0.035 2569.9 2567.7 0.086

889.39 883.58 0.658 1685.7 1679.7 0.357 2575.5 2574.6 0.035 2569.9 2567.7 0.086

1406.9 1398.6 0.594 2578.9 2578.7 0.008 2870.5 2883.0 +0.434 3611.6 3613.3 +0.047

1745.7 1736.2 0.547 2578.9 2578.7 0.008 2870.5 2883.0 +0.434 3619.3 3651.5 +0.882

1745.7 1736.2 0.547 2596.6 2592.5 0.158 3635.7 3633.3 0.066 3619.3 3651.5 +0.882

2244.9 2234.1 0.483 3170.7 3169.6 0.035 4181.8 4216.1 +0.814 5064.2 5057.5 0.133

2244.9 2234.1 0.483 3170.7 3169.6 0.035 4960.5 5010.6 +0.999 5064.2 5057.5 0.133

2579.9 2579.7 0.008 3645.5 3644.9 0.016 4960.5 5010.6 +0.999 5076.2 5135.9 +1.162

2579.9 2579.7 0.008 3989.9 3994.8 +0.123 5123.1 5115.7 0.145 5651.8 5624.3 0.489

Table 6 First 10 natural frequency (Hz) for SCSC Al/Al2O3 square plates (g = 1, p = 1).

s ¼ ha 0.05

0.1

0.2

0.3

Method

Present FEM Error (%) Present FEM Error (%) Present FEM Error (%) Present FEM Error (%)

Mode 1

2

3

4

5

6

7

8

9

10

522.04 519.30 0.527 991.09 988.89 0.223 1694.3 1699.3 +0.294 2138.4 2150.9 +0.581

979.44 973.88 0.571 1827.1 1823.0 0.225 2575.9 2574.6 0.051 2572.6 2567.7 0.191

1225.4 1220.1 0.434 2222.6 2220.0 0.117 3018.3 3033.8 +0.511 3728.9 3767.5 +1.025

1657.6 1650.1 0.454 2579.3 2578.7 0.023 3460.6 3456.8 0.110 4099.8 4072.2 0.678

1799.9 1790.9 0.502 2960.4 2958.0 0.081 4523.5 4531.3 +0.172 4502.6 4503.8 +0.027

2223.4 2217.0 0.289 3243.8 3244.1 +0.009 4525.1 4535.1 +0.221 4792.2 4773.0 0.402

2422.2 2412.3 0.410 3830.5 3840.6 +0.263 4823.4 4818.9 0.093 5076.9 5057.5 0.384

2581.5 2579.7 0.070 4217.0 4221.1 +0.097 5020.2 5070.6 +0.994 5341.4 5340.0 0.026

2642.5 2635.1 0.281 4495.3 4509.8 +0.322 5123.3 5115.7 0.149 5946.9 5958.7 +0.198

2930.7 2920.0 0.366 4540.5 4546.6 +0.134 5500.4 5539.9 +0.713 5946.9 6029.7 +1.373

Table 7 First 10 natural frequency (Hz) for SFSF Al/Al2O3 square plates (g = 1, p = 1).

s ¼ ha 0.05

0.1

0.2

0.3

Method

Present FEM Error (%) Present FEM Error (%) Present FEM Error (%) Present FEM Error (%)

Mode 1

2

3

4

5

6

7

8

9

10

175.32 174.82 0.286 348.01 345.38 0.761 664.95 661.68 0.494 935.96 933.81 0.230

291.78 289.77 0.694 567.98 564.12 0.684 1046.5 1041.1 0.519 1414.8 1409.6 0.369

656.45 653.31 0.481 1251.6 1244.4 0.579 1953.1 1952.4 0.036 1955.4 1954.4 0.051

702.78 698.85 0.562 1345.7 1339.4 0.470 2167.5 2162.3 0.240 2568.5 2567.7 0.031

836.11 832.72 0.407 1585.8 1578.3 0.475 2344.1 2347.0 +0.124 2757.7 2756.0 0.062

1246.6 1243.6 0.241 1952.3 1952.0 0.015 2574.2 2574.6 +0.015 3003.2 3021.4 +0.602

1328.9 1324.6 0.325 2308.5 2300.1 0.365 2699.1 2699.6 +0.019 3393.5 3400.6 +0.209

1554.3 1550.8 0.226 2455.8 2448.1 0.314 3645.3 3643.8 0.041 3653.1 3650.9 0.060

1688.0 1683.0 0.297 2578.7 2578.7 0 3727.1 3732.4 +0.142 4450.8 4432.8 0.406

1925.7 1921.4 0.224 2854.0 2849.5 0.158 3935.1 3945.1 +0.253 4474.0 4467.7 0.141

coupling (because of the existence of I22 and I4 terms). For simplicity, the authors name three models: First, UM as the uncoupled model when B; G, I22 and I4 terms are equal to zero. Second, PUM as the potential uncoupled model when B and G terms are assumed to be zero. Third, KUM as the kinetics uncoupled model when I22 and I4 terms are assumed to be zero. Fig. 3a–c shows the variation of the fundamental frequency parameters b of the Al/Al2O3 square plate (g = 1) for SSSC, SSSF and SCSF boundary conditions, respectively, when p = 1. In Fig. 3a–c, comparison between the results of different models (i.e. FEM, the present method, UM, PUM, KUM) is carried out. From Fig. 3a–c, it can be observed that the present results are in excellent agreement with those acquired by the FEM. It is also seen that

the KUM results are very close to those of the present and FEM results. This indicates that the kinetic coupling has a smaller effect on the frequency parameters. On the contrary, by considering of the PUM and UM results, it can be inferred that the frequency parameters are significantly influenced by the potential coupling parameters even if the plate is thin. On the other hand, from Eqs. (13a) and (13b), it can be concluded that the magnitude of the potential coupling parameters (i.e. B and G terms) and the kinetic coupling parameters (i.e. I22 and I4 terms) are strongly dependent on (Ec  Em) and (qc  qm), respectively. Hence, the difference between Young’s modulus of ceramic and metal in FGMs has a larger effect on the frequency parameters than the difference between their mass density.

732

Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

Fig. 3. Variation of the frequency parameters b of Al/Al2O3 square plates against thickness to length ratio s for (a) SSSC, (b) SSSF (c) SCSF when p = 1.

3.2.3. Effect of the power law index p on the eigenfrequency  parameters b Fig. 4a and b shows the variation of the eigenfrequency param versus the power law index p for a FG square plates with all eter b six possible combinations of boundary conditions when s = 0.15. It is seen that the power law index p has a highly significant influ Another interesting point ence on the eigenfrequency parameter b. attracting one’s attention is that regardless of the materials and  has the minboundary conditions, the eigenfrequency parameter b imum and maximum values at the points around p = 1 and p = 8, respectively. Herein, the minimum and maximum gradient index p denoted by pcrmin and pcrmax, respectively. Fig. 4a and b proves the fact that as the gradient index p varies from 103 to 103, the FG Al/Al2O3 material has a higher effect on the eigenfrequency param when compared with the FG Al/ZrO2 material. The reason is eter b

that, as mentioned in Section 3.2.2, the frequency parameters of FG plate are strongly dependent on (Ec  Em) and zirconia has a much smaller Young’s modulus than alumina. Due to the importance of the pcr value, Section 3.2.4 is devoted to determining the critical values of the gradient index p. 3.2.4. Variations of the critical power law index pcr with respect to thickness ratio s The graph of the minimum critical power law index pcrmin against the thickness to length ratio s is plotted in Fig. 5a for a square FG plates (g = 1) with SSSS, SCSC and SFSF boundary conditions. It can be seen that the values of the minimum critical gradient indices pcrmin for the Al/Al2O3 square plate are greater than those for the Al/ZrO2 one when any combinations of boundary conditions are taken into account. It is also observed in Fig. 5a that

Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

733

 versus the power law index p for a FG square plates with (a) SSSF, SCSF, SFSF and (b) SSSS, SSSC, SCSC boundary conditions Fig. 4. Variation of the eigenfrequency parameter b when s = 0.15.

Fig. 5. Variation of the (a) minimum critical gradient index pcr-min and (b) maximum critical gradient index pcr-max versus the thickness ratio s for a FG square plates with SSSS, SCSC and SFSF boundary conditions.

the pcrmin of the Al/ZrO2 square plate enhances with increasing the thickness ratio s whereas for the Al/Al2O3 plate the minimum critical power law index decreases. It is worthwhile to mention that the variation of the minimum critical gradient index pcrmin against the s is almost negligible for the Al/Al2O3 plates especially with SFSF and SSSS boundary conditions. A rapid decrease in the values of the minimum critical gradient index is observed for the SCSC Al/ Al2O3 square plate. In Fig. 5b, the influence of the thickness ratio s on the maximum power law index pcrmax is illustrated for a FG square plate for a FG square plate with SSSS, SCSC and SFSF boundary conditions. The primary conclusion inferred from Fig. 5b is to decrease the pcrmaxwith the increase the thickness ratio s, while all other parameters

are considered to be fixed. It is also concluded that the values of the pcrmax for the Al/ZrO2 plate are higher than those having the Al/ Al2O3 plates. It is evident from Fig. 5b that the SCSC boundary condition exerts a greater influence on the pcrmax in comparison with the SSSS and SFSF boundary conditions. 3.2.5. Effect of the aspect ratio g on the frequency parameters b The influence of the aspect ratio g = b/a on the frequency parameters b of a rectangular Al/ZrO2 plate (d = 0.2, p = 1) is shown in Table 8, for all six possible combinations of boundary conditions. From Table 8, it can be inferred that with the increase of the aspect ratio, the frequency parameter b decreases for all boundary conditions, while an inverse behavior is experienced for SFSF rectangular

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Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

Table 8 pffiffiffiffiffiffiffiffiffiffiffiffi Fundamental frequency parameter b ¼ xa2 qc =Ec =h for Al/ZrO2 plates (s = 0.2, p = 1).

g ¼ ba 0.5 2/3 1 1.5 2

Mode SSSS

SSSC

SCSC

SSSF

SCSF

SFSF

9.3216 7.5005 4.8909 3.6354 3.1796

12.871 8.9203 5.5570 3.8894 3.2965

15.178 10.409 6.3324 4.2025 3.4405

3.9492 3.4390 2.9918 2.7608 2.6749

5.1982 4.2807 3.1808 2.8196 2.6996

2.4820 2.4916 2.5098 2.5285 2.5395

plates. It is also found that the aspect ratio g exerts a greater influence on the frequency parameters of SSSS, SSSC and SCSC plates in comparison with the SSSF, SCSF and SFSF plates.

¼

1 ð48Fð12Gð1 þ mÞ þ s2 ð9Bð1 þ mÞ þ 4I4 b2 s2 ÞÞÞ y

þ s2 ð128HI4 b2 þ 3ð8I55 b2 þ 3Að1 þ mÞÞs2 ð4G þ 3Bs2 Þ þ 72Dð4Gð1 þ mÞ þ s2 ð3B  3Bm þ I4 b2 s2 ÞÞÞ; 1 y12 ¼ ð9A2 ð1 þ mÞs4 þ 8I4 s2 b2 ð4G  3Bs2 Þ þ 8Að18Fð1 þ mÞ y 16H ; 9s4 2 16I7 b ð1 þ mÞð16F  8Ds2 þ As4 Þ þ ; y15 ¼ I1 b2 s2 y y14 ¼ 2s 4 9s2 ¼ ð1 þ mÞð16G2  24GBs2 þ 9B2 s4 þ Að16H þ 24F s2  9Ds4 ÞÞ þ 3s2 ð3Dð1 þ mÞ þ I55 b2 s2 ÞÞÞ; y13 ¼

ðA:2Þ a1 ¼

1 að1 þ mÞ2

ð4ð2y13 þ ðy10 y8 þ y4 y9 Þð1 þ mÞÞÞ

ðA:3Þ

4. Conclusions This paper presents exact closed-form solutions for free vibration analysis of thick FG rectangular plates with different combinations of free, simply supported and clamped boundary conditions. Based on Reddy’s third-order shear deformation theory for FG plates, five highly coupled governing partial differential equations of motion for freely vibrating FG plates were exactly solved by introducing the potential functions and using the method of separation of variables. Several comparison cases were presented by those reported in the literature and the FEM analysis, for thin, moderately thick and thick plate to demonstrate highly stability and accuracy of present exact procedure. It was observed that the proposed exact method yields an improved accuracy for inplane and out-of-plane vibration of FG plates over the existing analytical and numerical methods based on TSDT in literature. Several numerical results are generated to manifest the effects of the material constant (i.e., the power-index g) and the aspect ratio on the natural frequencies. All analytical results presented here can be applied as a useful and reliable source by other research groups.

a2 ¼

1 ð2b2 s2 ð16HI1 þ 3s2 ð8FI1  4GI22 þ 3ðDI1 þ BI22 Þs2 ÞÞÞ; y2 y 1 4 4 ð6b2 s4 ð4GI1 þ 3s2 ðBI1 þ AI22 ÞÞÞ; y3 ¼ I4 b2 ; y4 ¼ 2 G; y5 ¼ y 3 3s 1 ¼ ð48Fð4Gð1 þ mÞ  s2 ð3Bð1 þ mÞ þ I22 b2 s4 ÞÞ y

y1 ¼

þ s2 ðs2 ð3Að1 þ mÞð4G  3Bs2 Þ þ 2b2 ð16HI22 þ 12GI33 s2  9BI33 s4 ÞÞ þ Dð96Gð1 þ mÞ þ 18s2 ð4Bð1 þ mÞ 1 ð3ð3A2 ð1 þ mÞs4 þ 2I22 b2 s4 ð4G  3Bs2 Þ þ I22 b2 s4 ÞÞÞÞ; y6 ¼ y þ 6Að8Fð1 þ mÞ þ s2 ð4D  4Dm þ I33 b2 s4 ÞÞÞÞ; y7 1 ¼ ð48Fð1 þ mÞ þ s2 ð24Dð1 þ mÞ þ ð3A þ 8I55 b2  3AmÞs2 ÞÞ y ðA:1Þ

ð2ð4y14 þ y1 ð2y13 þ y10 y8 ð1 þ mÞÞð1 þ mÞ

þ ð1 þ mÞð2y11 y4 þ y10 y4 y5 þ 2y13 þ 2y10 y7 þ 2y12 y8 þ 2y3 y9  y4 y6 y9 þ y2 y8 y9  my10 y4 y5 þ my4 y6 y9  my2 y8 y9 ÞÞÞ a3 ¼

1 að1 þ mÞ2

ðA:4Þ

ð2ð4y15 þ y1 ð2y14 þ ðy13 y6 þ y10 y7 þ y12 y8 Þ

 ð1 þ mÞÞð1 þ mÞ þ ð1 þ mÞðy10 y3 y5 þ y12 y4 y5 þ 2y14 y6 þ 2y12 y7  y3 y6 y9 þ y2 y7 y9 þ y11 ð2y3 þ ðy4 y6  y2 y8 Þð1 þ mÞÞ  y13 y2 y5 ð1 þ mÞ  my10 y3 y5  my12 y4 y5 þ my3 y6 y9  my2 y7 y9 ÞÞÞ a4 ¼

ðA:5Þ

2 ðy y y þ 2y15 y6  y11 y3 y6 þ y11 y2 y7 að1 þ mÞ 12 3 5 þ y1 ð2y15 þ ðy14 y6 þ y12 y7 Þð1 þ mÞÞ  y14 y2 y5 ð1

Appendix A Some coefficients referred to in this paper are given as follows

1 að1 þ mÞ2

a5 ¼

þ mÞ  my12 y3 y5 þ my11 y3 y6  my11 y2 y7 Þ

ðA:6Þ

1 ðy ð2y2 y5 þ 2y1 y6 ÞÞ a 15

ðA:7Þ

a ¼ 2y11  ðy10 y5 þ y1 y9 Þð1 þ mÞ

ðA:8Þ

  di ¼  2a6i ð2y12  ðy10 y6 þ y2 y9 Þð1 þ mÞÞð2y13 þ ðy10 y8 þ y4 y9 Þ  ð1 þ mÞÞÞ=ð1 þ mÞ  2y15 ð2y11 y2  2y12 y6  y1 y2 y9    y2 y6 y9 þ y26 þ y2 y5 y10 ð1 þ mÞ þ my1 y2 y9 þ my2 y6 y9 Þ 1  2 2 a 4y12 y7 ð1 þ mÞ þ ð2y11 ð2y14 y2  y10 ðy3 y6  1 þ m i  y2 y7 Þð1 þ mÞÞ þ y2 y9 ð4y15  2y1 y14 ð1 þ mÞ  ð2y14 y6 þ ðy3 y6  y2 y7 Þy9 ð1 þ mÞÞð1 þ mÞÞ   þ y10 ð4y15 y6  2 y26 þ y2 y5 y14 ð1 þ mÞ þ ðy1  y6 Þðy3 y6  y2 y7 Þy9 ð1 þ mÞ2 Þ þ y210 y5 ðy3 y6  y2 y7 Þð1 þ mÞ2 Þð1 þ mÞ þ 2y12 ð4y15 þ ð1 þ mÞð2y11 y3 þ 2y14 y6 þ y1 y3 y9  y3 y6 y9 þ 2y2 y7 y9

4ð4H þ 3F s2 Þ ; y8 ¼ 9s 4 1 y9 ¼ ð8ð4FG þ 4BH  3BF s2 þ 3DGs2 ÞÞ; y10 y 1 ¼ ð8ð4G2  4AH þ 3AF s2  3BGs2 ÞÞ; y11 y

 ðy6 y7 þ y3 y5 Þy10 ð1 þ mÞ  my1 y3 y9 þ my3 y6 y9  2my2 y7 y9 ÞÞÞÞ 1  4 2 a 4y12 y8 ð1 þ mÞ þ 2y12 ð4y14 þ ð1 þ mÞð2y11 y4 þ 1 þ m i þ 2y13 y6 þ 2y3 y9 þ y1 y4 y9  y4 y6 y9 þ 2y2 y8 y9  my1 y4 y9 þ my4 y6 y9  2my2 y8 y9 þ ð2y7 þ y4 y5 þ y6 y8  my4 y5  my6 y8 ÞÞÞ

Sh. Hosseini-Hashemi et al. / Composite Structures 93 (2011) 722–735

þ ð1 þ mÞð2y11 ð2y13 y2  ðy4 y6  y2 y8 Þy10 ð1 þ mÞÞ þ y2 y9 ð4y14  2y1 y13 ð1 þ mÞ  ð2y13 y6 þ y9 ð2y3 þ ðy4 y6  y2 y8 Þð1 þ mÞÞÞð1 þ mÞÞ þ y210 ð2y6 y7 þ y4 y5 y6 ð1 þ mÞ  y2 y5 y8 ð1 þ mÞÞð1 þ mÞ    þ y10 4y14 y6  2 y26 þ y2 y5 y13 ð1 þ mÞ  ðy1  y6 Þð2y3 y6  y4 y26 þ 2y2 y7 þ y2 y6 y8  ðy4 y6  y2 y8 Þy1 ð1 þ mÞ þ y4 y26 m  my2 y6 y8 ÞÞÞÞÞÞ=ð4y211 y2 þ y1 ð2y11 ð2y12  ðy10 y6  y2 y9 Þð1 þ mÞÞ  ðy6 y9 ð2y12  y2 y9 ð1 þ mÞÞ þ y10 ð2y12 y5    y26 þ y2 y5 y9 ð1 þ mÞÞ þ y6 y5 y210 ð1 þ mÞÞð1 þ mÞÞ  þ y5 ð4y212 þ 2y12 ðy10 y6 þ 2y2 y9 Þð1 þ mÞ  y2 y210 y5  þ y10 y9 y6 þ y2 y29 ð1 þ mÞ2 Þ þ y9 y21 ð2y12  y10 y6 ð1 þ mÞÞð1 þ mÞ þ y11 ð2y10 ðy26 þ 2y2 y5 Þð1 þ mÞ þ 2y6 ð2y12 þ y2 y9  y2 y9 mÞÞÞ; ði ¼ 1; 2; 3; 4Þ

ðA:9Þ

 diþ4 ¼ ð2y11  ðy10 y5 þ y1 y9 Þð1 þ mÞÞ 4a6i ð2y11  ðy10 y5 þ y1 y9 Þð1 þ mÞÞð2y13 þ ðy10 y8 þ y4 y9 Þð1 þ mÞÞ þ 4y15 ðy1 ð2y11 þ y10 y5 ð1 þ mÞÞ þ y5 ð2y12 þ ðy10 y6 þ y2 y9 Þð1 þ mÞÞ þ y21 y9 ð1 þ mÞÞð1 þ mÞ þ a2i ð2y12  ðy10 y6 þ y2 y9 Þð1 þ mÞÞð2ð2y14 y5 þ 2y11 y7 þ ðy3 y5  y1 y7 Þy9 ð1 þ mÞÞð1 þ mÞ þ ð2ð2y11  ðy10 y5 þ y1 y9 Þð1 þ mÞÞð4y15 þ y1 ð2y14 þ y10 y7 ð1 þ mÞÞð1 þ mÞ þ y3 ð1 þ mÞð2y11 þ y10 y5  my10 y5 ÞÞÞ=ð2y12  ðy10 y6 þ y2 y9 Þð1 þ mÞÞÞ  2a4i ð2y12  ðy10 y6 þ y2 y9 Þð1 þ mÞÞðð2y13 y5 þ 2y11 y8 þ ðy4 y5  y1 y8 Þy9 ð1 þ mÞÞð1 þ mÞ þ ðð2y11  ðy10 y5 þ y1 y9 Þð1 þ mÞÞð4y14 þ y1 ð2y13 þ y10 y8 ð1 þ mÞÞð1 þ mÞ þ ð1 þ mÞð2y11 y4 þ y10 y4 y5 þ 2y10 y7 þ 2y3 y9  my10 y4 y5 ÞÞÞ=ðð2y12  ðy10 y6 þ y2 y9 Þð1 þ mÞÞÞÞÞ=ð2ð2y11  ðy10 y5 þ y1 y9 Þð1 þ mÞÞð4y211 y2

  þ 2y11 ðy6 ð2y12 þ y2 y9 ð1 þ mÞÞ þ y10 2y2 y5 þ y26 ð1 þ mÞÞ þ y1 ð2y11 ð2y12  ðy10 y6  y2 y9 Þð1 þ mÞÞ þ ðy6 y9 ð2y12  y2 y9 ð1 þ mÞÞ þ y10 ð2y12 y5  ðy26 þ y2 y5 Þy9 ð1 þ mÞÞ þ y6 y5 y210 ð1 þ mÞÞð1 þ mÞÞ  þ y5 4y212  2y12 ðy10 y6 þ 2y2 y9 Þð1 þ mÞ   þ y2 y210 y5 þ y10 y9 y6 þ y2 y29 ð1 þ mÞ2 þ y9 y21 ð2y12 þ y10 y6 ð1 þ mÞÞð1 þ mÞÞð1 þ mÞÞ; ði ¼ 1; 2; 3; 4Þ

ðA:10Þ

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