Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate

ARTICLE IN PRESS JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 312 (2008) 862–892 www.elsevier.com/locate/jsvi

Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate Y.X. Haoa,b, L.H. Chena, W. Zhanga,, J.G. Leib a

College of Mechanical Engineering, Beijing University of Technology, Beijing 100022, PR China b Beijing Institute of Machinery, Beijing 100085, PR China

Received 12 August 2007; received in revised form 21 September 2007; accepted 11 November 2007

Abstract An analysis on the nonlinear dynamics of a simply supported functionally graded materials (FGMs) rectangular plate subjected to the transversal and in-plane excitations is presented in a thermal environment for the first time. Material properties are assumed to be temperature dependent. Based on Reddy’s third-order plate theory, the nonlinear governing equations of motion for the FGM plates are derived using Hamilton’s principle. Galerkin’s method is utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined parametric and external excitations. The resonant case considered here is 1:1 internal resonance and principal parametric resonance. The asymptotic perturbation method is utilized to obtain four-dimensional nonlinear averaged equation. The numerical method is used to find the nonlinear dynamic responses of the FGM rectangular plate. It was found that periodic, quasi-periodic solutions and chaotic motions exist for the FGM rectangular plates under certain conditions. It is believed that the forcing excitations f1 and f2 can change the form of motions for the FGM rectangular plate. r 2007 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGMs) are new engineering composite materials, which are being widely applied to large space station, shuttle, aircraft, automotive and many others in recent years [1,2]. The FGMs are microscopically inhomogeneous composites usually made from a mixture of metals and ceramics. By gradually varying the volume fraction of constituent materials, their material properties exhibit a smooth and continuous change from one surface to another. Thus, interface problems and mitigating thermal stress concentrations can be eliminated. In the FGMs, the micro-structures are spatially varied through non-uniform distribution of the reinforcement phases using the reinforcement with different properties, sizes and shapes as well as by interchanging the roles of the reinforcement and matrix phases in a continuous manner [3]. With the increasing use of FGM plates in engineering fields, research on the nonlinear dynamics, bifurcations, and

Corresponding author.

E-mail addresses: [email protected] (Y.X. Hao), [email protected] (W. Zhang). 0022-460X/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2007.11.033

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chaos of the FGM plates plays a significant role in applications. However, to date, only few studies on the bifurcations and chaos of the FGM plates have been conducted. In the last 10 years, several researchers have focussed their attention on investigating the dynamics the FGM plates. Praveen and Reddy [4] provided the nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates subjected to pressure loading and thickness varying temperature fields. Cheng and Batra [5] used Reddy’s third-order plate theory to study buckling and steady-state vibrations of a simply supported functionally gradient isotropic polygonal plate resting on a Winkler Pasternak elastic foundation and subjected to uniform in-plane hydrostatic loads. Ng et al. [6] analyzed the parametric resonance of functionally graded rectangular plates under harmonic in-plane loading. Yang and Shen [7] studied the dynamic responses of initially stressed FGM rectangular thin plates subjected to partially distributed impulsive lateral loads and without or resting on an elastic foundation. In 2002, they [8] investigated the free and forced vibrations of FGM plates in a thermal environment. The material properties were assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. Based on Reddy’s higher-order shear deformation shell theory, Yang and Shen [9] analyzed the free vibration and parametric resonance of shear deformable functionally graded cylindrical panels subjected to combined static and periodic axial forces in a thermal environment. Yang et al. [10] investigated the large amplitude vibration of pre-stressed FGM laminated plates that are composed of a shear deformable functionally graded layer and two surface-mounted piezoelectric actuator layers. In addition, Qian et al. [11] analyzed the static deformations, free and forced vibrations of a thick functionally graded elastic rectangular plate by using a higher-order shear and normal deformable plate theory and the Petrov-Galerkin method. Senthil and Batra [12] gave a three-dimensional exact solution for free and forced vibrations of simply supported FGMs rectangular plates. Huang and Shen [13] studied the nonlinear vibrations and dynamic responses of FGM plates in thermal environment, in which the heat conduction and temperature-dependent material properties were both considered. Chen [14] investigated the nonlinear vibrations of FGM plates with arbitrary initial stresses. The effect of transverse shear strains and rotary inertia was included in five partial differential governing equations. Studies of the nonlinear vibrations and dynamic stability of the plate and shells have been extensively conducted in the past two decades. Many of these studies are focused on isotropic or laminated composite plates and shells. Hadian and Nayfeh [15] used the method of multiple scales to analyze asymmetric nonlinear responses of clamped circular plates subjected to harmonic excitations and considered the case of a combination-type internal resonance. Chang et al. [16] investigated the bifurcations and chaos of a rectangular thin plate with 1:1 internal resonance. Anlas and Elbeyli [17] studied the nonlinear dynamics of a simply supported rectangular plate subjected to transverse harmonic excitation. Zhang et al. [18] investigated the global bifurcations and chaotic dynamics of a parametrically and externally excited simply supported rectangular thin plate. Recently, Ye et al. [19] studied the local and global nonlinear dynamics of a parametrically excited rectangular symmetric cross-ply laminated composite plate. In addition, Zhang et al. [20] gave further studies on the nonlinear oscillations and chaos of a rectangular symmetric cross-by laminated plate under parametric excitation. In the last century, to investigate the nonlinear oscillations, many asymptotic perturbation techniques, such as the averaging method, the Krylov, Bogoliubov and Mitropolsky (KBM) method, the method of multiple scales and the harmonic balance method, have been presented and widely used to construct the approximate solutions of weakly nonlinear systems. In general situations, analysis is carried out only up to the first-order approximation since higher-order terms do not have large influence on the qualitative characteristics of the asymptotic solutions. However, the quadratic nonlinearities cannot be included in the first-order approximate solutions when they exist in the original nonlinear systems. Therefore, in order to acquire better qualitative and quantitative characteristics of nonlinear systems having the quadratic and cubic nonlinearities, the second-order averaging or perturbation procedure should be considered. In order to investigate conveniently nonlinear dynamic responses of systems including the quadratic and cubic nonlinearities, an asymptotic perturbation method was developed by Maccari [21–25] based on the slow temporal rescaling and balancing of the harmonic terms with a simple iteration. In the certain sense, the asymptotic perturbation method can be considered as an attempt to link the most useful characteristics of the harmonic balance and the method of

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multiple scales. Recently, Ye et al. [26] utilized the asymptotic perturbation method to study the nonlinear oscillations and chaotic dynamics of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation. This paper focuses on research on the bifurcations and chaotic dynamics of a simply supported at the four edges, FGM rectangular plate subjected to the in-plane and transversal excitations simultaneously in the uniform thermal environment. Material properties of the constituents are graded in the thickness direction according to a power-law distribution. In the framework of Reddy’s third-order shear deformation plate theory [27], the governing equations of motion for the FGMs rectangular plate are derived using Hamilton’s principle. Because only transverse nonlinear oscillations of the FGM plate are considered, the equations of motion can be reduced into a two-degree-of-freedom nonlinear system under combined parametric and external excitations using Galerkin’s method. The case of 1:1 internal resonance and primary parametric resonance is considered for the FGM rectangular plate. The asymptotic perturbation method developed by Maccari [21–25] is employed to transfer the second-order nonautonomous nonlinear differential equation with the quadratic and cubic nonlinear terms to the first-order nonlinear averaged equation. Using the numerical method, the averaged equation is analyzed to find the nonlinear responses and chaotic motions of the FGM rectangular plate. 2. Formulation A simply supported at the four-edges FGMs rectangular plate subjected to the in-plane and transversal excitations is considered, as shown in Fig. 1. The edge width and length of the FGM rectangular plate in the x and y directions are, respectively, a and b and the thickness is h. A Cartesian coordinate Oxyz is located in the middle surface of the FGMs rectangular plate. Assume that (u, v, w) and (u0, v0, w0) represent the displacements of an arbitrary point and a point in the middle surface of the FGMs rectangular plate in the x, y and z directions, respectively. It is also assumed that fx and fy, respectively, represent the mid-plane rotations of two transverse normals about the x-and y-axis. The in-plane excitation of the FGMs plate is distributed along the y direction at x ¼ 0 and a and is of the form (p0p1 cos O2t). The transversal excitation subject to the FGMs plate is represented by F(x,y)cos O1t. Here, O1 and O2 are the frequencies of the transversal excitation and the in-plane excitation, respectively. 2.1. Materials properties Generally speaking, most of the FGMs are employed in high-temperature environment and many of the constituent materials may possess temperature-dependent properties. It is assumed that the plate is made from a mixture of ceramics and metals with continuous varying such that the bottom surface of the plate is metalrich, whereas the top surface is cerami-rich. The material properties P, such as Young’s modulus E and the coefficient a of thermal expansion, can be expressed as a function of the temperature [28,29] Pi ¼ P0 ðP1 T 1 þ 1 þ P1 T þ P2 T 2 þ P3 T 3 Þ, where P0, P1, P1, P2 and P3 are temperature coefficients.

Fig. 1. The model of an FGM rectangular plate and the coordinate system.

(1)

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The effective material properties P of the FGMs can be expressed as P ¼ Pt V c þ Pb V m ,

(2)

where subscripts t and b, respectively, represent the top and bottom surfaces of the FGMs plate, Vc and Vm are the ceramic and metal volume fractions and add to unity V c þ V m ¼ 1.

(3)

The metal volume fraction Vm is defined as [37]  V m ðzÞ ¼

2z þ h 2h

N ,

(4)

where power-law exponent N is a real number that characterizes the metal variation profile through the plate thickness. From Eqs. (2)–(4), Young’s modulus E, the coefficient a of the thermal expansion, the mass density r and the thermal conductivity k can be expressed as E ¼ ðE b  E t ÞV m þ E t ,

(5a)

a ¼ ðab  at ÞV m þ at ,

(5b)

r ¼ ðrb  rt ÞV m þ rt ,

(5c)

k ¼ ðkb  kt ÞV m þ kt .

(5d)

We assume that the temperature variation occurs in the thickness direction only and the one-dimensional temperature field is constant in the xy plane of the plate. In this case, the temperature distribution along the thickness of the plate can be obtained by solving a steady-state heat transfer equation   d dT  kðzÞ ¼ 0. (6) dz dz This equation is solved by imposing the boundary condition of T ¼ Tt at z ¼ h/2 and T ¼ Tb at z ¼ h/2. For an isotropic material, the solution of Eq. (6) may be expressed as Tt þ Tb Tt  Tb þ z. (7) 2 h It is also supposed that the FGMs plate is linear elastic throughout the deformation, and that the plate is initially stress-free at T0 and is subjected to a uniform temperature variation DT ¼ TT0. TðzÞ ¼

2.2. Equations of motion According to Reddy’s third-order shear deformation (TSDT) in Refs. [27,30–32], the displacement field of the FGMs plate is assumed to be   qw0 uðx; y; tÞ ¼ u0 ðx; y; tÞ þ zfx ðx; y; tÞ  c1 z3 fx þ , qx   qw0 , vðx; y; tÞ ¼ v0 ðx; y; tÞ þ zfy ðx; y; tÞ  c1 z3 fy þ qy wðx; y; tÞ ¼ w0 ðx; y; tÞ. Based on the nonlinear strain–displacement relation and the above displacement field, we obtain       qu 1 qw 2 qv 1 qw 2 1 qu qv qw qw þ þ þ þ ; yy ¼ ; gxy ¼ , xx ¼ qx 2 qx qy 2 qy 2 qx qy qx qy     1 qv qw 1 qu qw þ þ ; gzx ¼ , gyz ¼ 2 qz qy 2 qz qx

ð8Þ

ð9Þ

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and 8 ð1Þ 9 8 ð3Þ 9 9 8 ð0Þ 9 8 xx > xx >  > ( > > ) ( ð0Þ ) ( ð2Þ ) xx > > > > > > > > = = = = < < < xx > < gyz gyz gyz ð0Þ ð1Þ ð3Þ 3 2 yy ¼ yy þ z yy þ z yy ; ¼ þz , ð0Þ gzx > > > > > > gzx gð2Þ ; > > > > :g > zx ; ; ; : gð0Þ > : gð1Þ > : gð3Þ > xy

xy

xy

(10)

xy

where 8 > > > > > > > > > <

  qu0 1 qw0 2 þ qx 2 qx   qv0 1 qw0 2 þ qy 2 qy

9 > > > > > > > > > =

9 8 qfx > > > > 8 ð0Þ 9 8 ð1Þ 9 > > > > qx > >   > > > > > > xx > xx > > > > > > > > > > > = = = < < < qf y ð0Þ ð1Þ yy yy ; , ¼ ¼ c1 qy > > > > > > > > > > > > > > ð0Þ > > > > > > > > > > > > ; > : gxy ; > > : gð1Þ > > xy > > > qfx qfy > > > > > qu0 qv0 qw0 qw0 > > > > > þ > ; > : þ þ ; : qy qx qy qx qx qy 9 8 qfx q2 w0 > > > > > > þ 8 ð3Þ 9 > > 2 > > qx qx > >  > > > > > > > > xx > > > > 2 = < = < qfy q w0 ð3Þ yy þ , ¼ c1 qy qy2 > > > > > > > > > > > > ð3Þ > > : gxy ; > > 2 > > > > > qfx þ qfy þ 2 q w0 > > > : qy qx qx qy ; 9 9 8 8 qw0 > 8 qw0 > 8 9 > 9 > > > > > þ f þ f ð0Þ ð2Þ > > > > < y < gyz = < y qy = < gyz = qy = 4 ; ; c2 ¼ 3c1 ; c1 ¼ h2 . ¼ ¼ c2 ð2Þ > > > : gð0Þ ; > : ; 3 qw0 > qw0 > gzx > > zx > > > > ; ; : fx þ : fx þ qx qx Taking into account the thermal effects, the stress–strain relationship is as follows: 9 8 988 9 8 9 9 8 sxx > > Q11 Q12 0 0 0 >>> xx > > axx > > > > > > > > > > > > > > > >> > > > > > > > > > > > > > > > > > > > > > > yy > syy > ayy > Q21 Q22 0 0 0 > > > > > > > > > > = = < =<< = < = > < gyz  syz ¼ 0 0 0 Q44 0 0 DT , > > >> > > > > > > > > > > > > > > > > > > > > > > gzx > 0 > szx > 0 0 0 Q55 0 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ; : ; > ; ; ; :s > : 0 : : g 2a 0 0 0 Q66 xy xy xy

ð11Þ

(12)

where the elastic stiffness coefficients of the FGM plate are given by Q11 ¼ Q22 ¼

E ; 1  n2

Q12 ¼ Q21 ¼

nE ; 1  n2

Q44 ¼ Q55 ¼ Q66 ¼

E 2ð1 þ nÞ

(13)

and the coefficient of thermal expansion can be written as axx ¼ ayy ¼ a; axy ¼ 0.

(14)

According to the Hamilton’s principle, the nonlinear governing equations of motion for the FGM rectangular plate are given as €  c1 I 3 qw€ 0 , N xx;x þ N xy;y ¼ I 0 u€ 0 þ ðI 1  c1 I 3 Þf x qx

(15a)

qw€ 0 , qy

(15b)

N yy;y þ N xy;x ¼ I 0 v€0 þ ðI 1  c1 I 3 Þf€ y  c1 I 3

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qw0 q2 w0 qw0 qw0 q2 w 0 qw0 q2 w0 þ N yy þ N þ 2N þ N þ N þ N xy;x xy;y xy xx;x xx qy qy2 qy qx qy qx qx qx2 þ c1 ðPxx;xx þ 2Pxy;xy þ Pyy;yy Þ þ ðQx;x  c2 Rx;x Þ þ ðQy;y  c2 Ry;y Þ þ F  gw_ 0 !  2    € € qf qu€ 0 q€v0 qf q w€ 0 q2 w€ 0 y x 2 þ þ  c1 I 6 ¼ I 0 w€ 0 þ c1 I 3 þ , þ c1 ðI 4  c1 I 6 Þ qx qx qx qy qx2 qy2

867

N yy;y

€ M xx;x þ M xy;y  c1 Pxx;x  c1 Pxy;y  ðQx  c2 Rx Þ ¼ ðI 1  c1 I 3 Þu€ 0 þ ðI 2  2c1 I 4 þ c21 I 6 Þf x qw€ 0 ,  c1 ðI 4  c1 I 6 Þ qx € M yy;y þ M xy;x  c1 Pyy;y  c1 Pxy;x  ðQy  c2 Ry Þ ¼ ðI 1  c1 I 3 Þ€v0 þ ðI 2  2c1 I 4 þ c21 I 6 Þf y qw€ 0 ,  c1 ðI 4  c1 I 6 Þ qy

ð15cÞ

ð15dÞ

ð15eÞ

where g is the damping coefficient, a comma denotes the partial differentiation with respect to a specified coordinate, a super dot implies the partial differentiation with respect to time, the stress resultants are represented as follows: 9 9 8 8 ð0Þ 9 8 T 9 8 8 ð0Þ 9 8 T 9 N xx > M xx > > > M xx > N xx >   > > > > > > > > > > > < > > > < > > > > > > > > = = = = = = < < < < T T ð1Þ ð1Þ N yy ¼ f½A½B½Eg  M yy ¼ f½B½D½F g  þ N yy ; þ M yy , > > > > > > > > > > > > > > > > > > > ; ; > ; > > > :N > : ð3Þ > : ð3Þ > ; : M xy ; ; : N Txy > : M Txy > xy   9 8 9 8 8 ð0Þ 9 T > Pxx >  > > ( > Pxx > > > ) ( ð0Þ ) ( ) ( ð0Þ ) > > > > = = = > < T > < < Qy Ry g g Pyy ¼ f½E½F ½Hg ð1Þ þ Pyy ; ¼ f½A½Dg ð2Þ ; ¼ f½D½F g ð2Þ , ð16Þ > > > > > > Qx Rx g g > > > ; ; > > PT > :P > : ð3Þ > ; : xy  xy where Aij, Bij, Dij, Eij, Fij and Hij, respectively, are the stiffness elements of the FGM plate, which are denoted as Z

h=2

Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz

ðAij ; Bij ; Dij ; E ij ; F ij ; H ij Þ ¼

ði; j ¼ 1; 2; 6Þ,

(17)

h=2

Z

h=2

Qij ð1; z2 ; z4 Þ dz

ðAij ; Dij ; F ij Þ ¼

ði; j ¼ 4; 5Þ

h=2

and the thermal stress resultants in Eq. (16) can be represented as 8 T 9 2 38 9 N xx > > Q11 Q12 0 >a> > > > > > > Z h=2 < T = 6 7< = N yy ¼  6 Q21 Q22 7 0 4 5> a >DT dz, > > h=2 > > > > > > T : N xy ; 0 0 Q66 : 0 ; 8 T 9 2 38 9 M xx > > Q11 Q12 0 >a> > > > > > > Z < T = h=2 6 7< = M yy ¼  6 Q21 Q22 0 7 a zDT dz; 4 5> > > > h=2 > > > > > MT > ; : 0 0 Q66 : 0 ; xy

(18)

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8 T 9 2 Pxx > > Q11 > > > > Z < T = h=2 6 Pyy ¼  6 Q21 4 > > h=2 > > > > PT ; : 0 xy

Q12 Q22 0

38 9 > > >a> 7< = 3 0 7 a z DT dz. 5> > > > Q66 : 0 ; 0

ð19Þ

It is found that the N Txx and N Tyy in Eq. (19) are the functions with respect to a and DT. They represent the thermal stress resultants, which demonstrate the thermal effect. Substituting the stress resultants of Eq. (16) into Eq. (15), we can write Eq. (15) in terms of generalized displacements ðu0 ; v0 ; w0 ; fx ; fy Þ as A11

q2 u0 q2 u0 q2 v 0 q2 fx q2 fx þ ðB þ A þ ðA þ A Þ þ c E Þ þ ðB þ c E Þ 66 12 66 11 1 11 66 1 66 qx2 qy2 qx qy qx2 qy2

þ ðB12  c1 E 12 þ B66  c1 E 66 Þ

q3 w0 q3 w0 €  c1 I 3 qw€ 0 ,  c1 ðE 12 þ 2E 66 Þ ¼ I 0 u€ 0 þ ðI 1  c1 I 3 Þf x 3 qx qx qy2 qx

 c1 E 11

A66

ð20aÞ

q2 fy q2 fy q2 v 0 q2 v 0 q2 u0 þ ðB þ A þ ðA þ A Þ þ c E Þ þ ðB þ c E Þ 22 21 66 66 1 66 22 1 22 qx2 qy2 qx qy qx2 qy2

þ ðB12  c1 E 21 þ B66  c1 E 66 Þ  c1 E 22

A21

q2 fy qw0 q2 w0 qw0 q2 w0 qw0 q2 w0 þ A11 þ A þ ðA þ A Þ 66 12 66 qx qy qx qx2 qx qy2 qy qx qy

q2 fx qw0 q2 w0 qw0 q2 w0 qw0 q2 w0 þ A66 þ A þ ðA þ A Þ 22 21 66 qx qy qy qx2 qy qy2 qx qx qy

q3 w 0 q3 w 0 qw€ 0 ¼ I 0 v€0 þ ðI 1  c1 I 3 Þf€ y  c1 I 3 ,  c ðE þ 2E Þ 1 12 66 qy3 qx2 qy qy

ð20bÞ

qu0 q2 w0 qu0 q2 w0 qu0 q2 w0 q2 u0 qw0 q2 u0 qw0 q2 u0 qw0 þ ðA þ A þ A þ A þ 2A þ A Þ 11 66 21 66 11 66 qx qy2 qx qx2 qy qx qy qx qy qy qx2 qx qy2 qx

þ c1 E 11

q3 u0 q3 u0 qv0 q2 w0 qv0 q2 w0 qv0 q2 w0 q2 v0 qw0 þ A þ c ðE þ 2E Þ þ 2A þ A þ A 1 21 66 66 22 12 22 qx3 qx qy2 qx qx qy qy qy2 qy qx2 qy2 qy

q2 v0 qw0 q2 v0 qw0 q3 v 0 q3 v 0 q2 w0 2 þ ðA þ c þ A Þ E þ c ðE þ 2E Þ þ ðA þ c F  2c D Þ 12 66 1 22 1 12 66 55 55 2 55 2 qx2 qy qx qy qx qy3 qy qx2 qx2  2 q2 w 0 q2 w0 q2 w0 q2 w0 q2 w0 q2 w0 2 þ ðA44 þ c2 F 44  2c2 D44 Þ 2 þ 2c1 ðE 66  E 12 Þ 2 þ 2c1 E 66 þ 2c1 ðE 21  2E 66 Þ qy qx qy2 qx qy qy2 qx qy þ A66

þ 2ðA21 þ 2A66 Þ

        qw0 q2 w0 qw0 3 qw0 2 q2 w0 1 qw0 2 q2 w0 1 qw0 2 q2 w0 þ A22 þ þ A þ A þ A A 21 66 66 21 qx qx qy qy qy qy2 qx qy2 qy qx2 2 2 2

  3 qw0 2 q2 w0 q4 w0 q4 w0 q4 w 0 qf 2 2 2 þ A11  c H  c H  2c ðH  2H Þ þ ð2c2 D55 þ c22 F 55 þ A55 Þ x 21 66 1 11 1 22 1 2 qx qx2 qx4 qy4 qx2 qy2 qx þ ðB21  c1 E 21 Þ

qfx q2 w0 qfx q2 w0 þ 2ðB  c E Þ 66 1 66 qx qy2 qy qx qy

þ ðB21 þ B66  c1 E 21  c1 E 66 Þ

q2 fx qw0 q2 f qw0 qf q2 w0 þ ðB11  c1 E 11 Þ 2x þ ðB11  c1 E 11 Þ x qx qy qy qx qx qx qx2

q2 fx qw0 q3 fx q3 f x 2 þ ðc F  c H Þ þ c ðF þ 2F  c H  2c H Þ 1 11 11 1 21 66 1 21 1 66 1 qy2 qx qx3 qx qy2 2 qfy qfy q w0 þ ðB22  c1 E 22 Þ þ ð2c2 D44 þ A44 þ c22 F 44 Þ qy qy qy2 þ ðB66  c1 E 66 Þ

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þ 2ðB66  c1 E 66 Þ þ ðB22  c1 E 22 Þ

869

qfy q2 w0 q2 fy qw0 qfy q2 w0 þ ðB66  c1 E 66 Þ 2 þ ðB12  c1 E 12 Þ qx qx qy qx qy qy qx2

q2 fy qw0 q2 fy qw0 q3 fy 2 þ ðB þ ðc þ B  c E  c E Þ F  c H Þ 12 66 1 12 1 66 1 22 22 1 qy2 qy qx qy qx qy3

q3 fy q2 w0 q2 w 0  N Txx  N Tyy þ F cos O1 t 2 2 qx qy qx qy2 !  2    qw0 q w€ 0 q2 w€ 0 qu€ 0 q€v0 qf€ x qf€ y 2 ¼ I 0 w€ 0  c1 I 6 þ þ þ g þ c1 I 3 þ c1 ðI 4  c1 I 6 Þ , qt qx2 qy2 qx qy qx qy

þ c1 ðF 12 þ 2F 66  c1 H 12  2c1 H 66 Þ

ðB11  c1 E 11 Þ

ð20cÞ

q2 u0 q2 u0 q2 v 0 qw0 q2 w0 þ ðB11  c1 E 11 Þ þ ðB66  c1 E 66 Þ 2 þ ðB12 þ B66  c1 E 12  c1 E 66 Þ 2 qx qy qx qy qx qx2

þ ðB12 þ B66  c1 E 12  c1 E 66 Þ

qw0 q2 w0 qw0 q2 w0 q3 w 0 2 þ ðB66  c1 E 66 Þ þ ðc F þ c H Þ 1 11 11 1 qy qx qy qx qy2 qx3

þ c1 ðF 12  2F 66 þ c1 H 12 þ 2c1 H 66 Þ

q3 w0 qw0 q2 f x 2 2 þ ðD  ðA  2c D þ c F Þ  2c F þ c H Þ 55 2 55 55 11 1 11 11 2 1 qx qy2 qx qx2

q2 f y q2 f x 2 2 þ ðD  2c F  2c F þ D þ c H þ c H Þ 12 1 12 1 66 66 1 12 1 66 qy2 qx qy qw€ 0 ,  ðA55  2c2 D55 þ c22 F 55 Þfx ¼ ðI 1  c1 I 3 Þu€ 0 þ ðI 2  2c1 I 4 þ c21 I 6 Þf€ x  c1 ðI 4  c1 I 6 Þ qx

þ ðD66  2c1 F 66 þ c21 H 66 Þ

ðB66  c1 E 66 Þ

ð20dÞ

q2 v 0 q2 v 0 q2 u0 qw0 q2 w0 þ ðB66  c1 E 66 Þ þ ðB22  c1 E 22 Þ 2 þ ðB21 þ B66  c1 E 21  c1 E 66 Þ 2 qx qy qx qy qy qx2

þ ðB22  c1 E 22 Þ

qw0 q2 w0 qw0 q2 w0 q3 w 0 þ ðc1 F 22 þ c21 H 22 Þ 3 þ ðB21 þ B66  c1 E 21  c1 E 66 Þ 2 qy qy qx qx qy qy

þ c1 ðF 21  2F 66 þ c1 H 21 þ 2c1 H 66 Þ

q2 f y q3 w0 qw0 2 2  ðA þ ðD  2c D þ c F Þ  2c F þ c H Þ 44 2 44 44 66 1 66 66 2 1 qx2 qy qy qx2

q2 f y q2 fx þ ðD21  2c1 F 21  2c1 F 66 þ D66 þ c21 H 21 þ c21 H 66 Þ 2 qy qx qy €  c1 ðI 4  c1 I 6 Þ qw€ 0 ,  ðA44  2c2 D44 þ c22 F 44 Þfy ¼ ðI 1  c1 I 3 Þ€v0 þ ðI 2  2c1 I 4 þ c21 I 6 Þf y qy

þ ðD22  2c1 F 22 þ c21 H 22 Þ

ð20eÞ

where all kinds of inertias in Eq. (20) are calculated by Z

h=2

zi rðzÞ dz

Ii ¼

ði ¼ 0; 1; 2; 3; 4; 6Þ.

(21)

h=2

The simply supported boundary conditions can be expressed as at x ¼ 0 and x ¼ a; w ¼ fy ¼ M xx ¼ Pxx ¼ N xy ¼ 0,

(22a)

at y ¼ 0 and y ¼ b; w ¼ fx ¼ M yy ¼ Pyy ¼ N xy ¼ 0,

(22b)

Z

Z

b

N yy jy¼0; b ¼ 0;

N xx jx¼0;a dy ¼  0

b

ðp0  p1 cos O2 tÞ dy.

(22c)

0

It is obvious that the boundary condition (22c) also includes the influence of the thermal environment.

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In order to obtain the dimensionless equations, we introduce the transformations of the variables and parameters 7=2 v0 w0 ¯ x ¼ fx ; f¯ y ¼ fy ; x¯ ¼ x ; y¯ ¼ y ; F¯ ¼ ðabÞ F , v¯ ¼ ; w ; f ¯ ¼ a b b h p4 Eh7     1=2 2 2 ðabÞ2 1 1=2 ¯ 0 ¼ b p0 ; P ¯ 1 ¼ b p1 ; t¯ ¼ p2 E g; P t, g¯ ¼ abr p2 h4 rE Eh3 Eh3  1=2 1=2 1=2 1=2 ¯ i ¼ 1 abr ¯ ij ¼ ðabÞ Aij ; B¯ ij ¼ ðabÞ Bij ; D ¯ ij ¼ ðabÞ Dij , O Oi ði ¼ 1; 2Þ; A 2 3 2 p E Eh Eh Eh4 1=2 1=2 ðabÞ1=2 1 ¯ ij ¼ ðabÞ F ij ; H ¯ ij ¼ ðabÞ H ij ; I¯ i ¼ I i. E ; F E¯ ij ¼ ij 5 6 8 ðiþ1Þ=2 Eh Eh Eh r ðabÞ

u¯ ¼

u0 ; a

ð23Þ

We mainly consider transverse nonlinear oscillations of the FGM rectangular plate in the first two modes. It is our aim to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the FGM rectangular plates. Thus, we write w ¯ as follows: px¯ 3p¯y 3px¯ p¯y sin þw sin , ¯ 2 ð¯tÞ sin a b a b where w ¯ 1 and w¯ 2 are the amplitudes of two modes, respectively. The transverse excitation can be represented as wð ¯ x; ¯ y¯ ; ¯tÞ ¼ w ¯ 1 ð¯tÞ sin

(24)

px¯ 3p¯y ¯ 3px¯ p¯y þ F 2 ð¯tÞ sin sin , (25) F¯ ðx; ¯ y¯ ; ¯tÞ ¼ F¯ 1 ð¯tÞ sin sin a b a b where F¯ 1 and F¯ 2 represent the amplitudes of the transverse forcing excitation corresponding to the two nonlinear modes. For simplicity, we drop the overbar in the following analysis. Based on research given in Refs. [33,34], neglecting all inertia terms on u, v, fx and fy in Eq. (20) and substituting Eq. (24) into Eqs. (20a), (20b), (20d) and (20e), we obtain the displacements u, v, fx and fy with respect to w. Substituting Eqs. (24) and (25) into Eq. (20c) and applying the Galerkin procedure yield the governing differential equation of transverse motion of the FGM rectangular plate for the dimensionless as follows: w€ 1 þ o21 w1 þ a1 w_ 1 þ a2 w1 cos O2 t þ a3 w21 þ a4 w22 þ a5 w1 w22 þ a6 w31 þ a7 w1 w2 ¼ f 1 cos O1 t,

(26a)

w€ 2 þ o22 w2 þ b1 w_ 2 þ b2 w2 cos O2 t þ b3 w1 w2 þ b4 w21 þ b5 w22 þ b6 w2 w21 þ b7 w32 ¼ f 2 cos O1 t,

(26b)

where all coefficients can be found in Appendix A, the f1 and f2 are the magnitudes of the forcing excitations, which are also given in Appendix A. The aforementioned equation, which includes the quadratic terms, cubic terms, parametric and transverse excitations, describes the nonlinear transverse vibrations of the FGM rectangular plate subjected to the inplane and transversal excitations in the first two modes. To consider the influence of the quadratic terms on the nonlinear dynamic characteristics of the FGM rectangular plate, we need to obtain the second-order approximate solution of Eq. (26). It is difficult for one to use the method of multiple scales to obtain the second-order approximate solution of Eq. (26). Thus, in the following analysis, we will utilize the asymptotic perturbation method presented by Maccari [21–25] to obtain the second-order approximate solution of Eq. (26). Utilizing the asymptotic perturbation method, we can obtain increasingly accurate solution by increasing the order of approximation in terms of the small parameter e. 3. Perturbation analysis To guarantee the validity of perturbation analysis, we use the asymptotic perturbation method [21–26,35] to obtain the averaged equation of system (26). It is assumed that the width-to-length ratio of the FGM rectangular plate is a/b ¼ 1. Therefore, we only consider the case of 1:1 internal resonance and primary parametric resonance for the FGM rectangular plate.

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In this resonant case, there are the following relations: o1 ¼

O1 O1 þ 2 s1 ; o2 ¼ þ 2 s2 ; O2 ¼ O1 ¼ O, 2 2

(27)

where o1 and o2 are two linear frequencies, s1 and s2 are the two detuning parameters. The scale transformations may be introduced as a1 !  2 a1 ; a2 !  2 a2 ; f 1 !  2 f 1 ; b1 !  2 b1 ; b2 !  2 b2 ; f 2 !  2 f 2 .

(28)

Substituting Eqs. (27) and (28) into Eq. (26) yields  2  O 4 2 2 þ  s1 þ Os1  w1 þ 2 a1 w_ 1 þ 2 a2 w1 cos Ot þ a3 w21 þ a4 w22 þ a5 w1 w22 þ a6 w31 þ a7 w1 w2 w€ 1 þ 4 ¼ 2 f 1 cos Ot,

ð29aÞ



 O2 4 2 2 þ  s2 þ Os2  w2 þ 2 b1 w_ 2 þ 2 b2 w2 cos Ot þ b3 w1 w2 þ b4 w21 þ b5 w22 þ b6 w2 w21 þ b7 w32 w€ 2 þ 4 ¼ 2 f 2 cos Ot,

ð29bÞ

We now introduce the temporal rescaling t ¼ q t,

(30)

where q is a rational positive number, which will be fixed afterwards. The value of q fixes the magnitude order of the temporal asymptotic limit in such a way that the nonlinear effects become consistent and non-negligible. If t ! 1, we set  ! 0, so that t assumes a finite value. The approximate solutions w1(t) and w2(t) of Eq. (29) are sought in a power series of small parameter e w1 ðtÞ ¼

þ1 X

dn cn ðt; Þ einðO=2Þt ,

(31a)

dn Fn ðt; Þ einðO=2Þt ,

(31b)

n¼1

w2 ðtÞ ¼

þ1 X n¼1

where dn ¼ jnj for na0, and d0 ¼ d is a positive number, which will be fixed later on. Because w1(t) and w2(t) are real, we can obtain cn ðt; Þ ¼ cnn ðt; Þ,

(32a)

Fn ðt; Þ ¼ Fnn ðt; Þ,

(32b)

where the asterisk denotes the complex conjugate. Therefore, the assumed solution (31) can be rewritten more explicitly as follows: w1 ðtÞ ¼ d c0 ðt; Þ þ c1 ðt; Þ eiðO=2Þt þ 2 c2 ðt; Þ eiOt þ 3 c3 ðt; Þ eið3O=2Þt þ 4 c4 ðt; Þ ei2Ot þ cc þ Oð5 Þ,

ð33aÞ

w2 ðtÞ ¼ d F0 ðt; Þ þ F1 ðt; Þ eiðO=2Þt þ 2 F2 ðt; Þ eiOt þ 3 F3 ðt; Þ eið3O=2Þt þ 4 F4 ðt; Þ ei2Ot þ cc þ Oð5 Þ,

ð33bÞ

where the symbol cc stands for the parts of complex conjugate of the functions on the right-hand side of Eq. (33). It is seen from Eq. (33) that the solutions of Eq. (29) can be considered as a combination of the various harmonics with coefficients depending on t and e. Assume that the functions cn ðt; Þ and Fn ðt; Þ can be

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expanded in power series of small parameter e cn ðt; Þ ¼

þ1 X

i cðiÞ n ðtÞ,

(34a)

i FðiÞ n ðtÞ.

(34b)

i¼0

Fn ðt; Þ ¼

þ1 X i¼0

It is also supposed that the limits of functions cn ðt; Þ and Fn ðt; Þ as  ! 0 exist and are finite. For ð0Þ ð0Þ ð0Þ simplicity of analysis, we use abbreviations cð0Þ n ¼ cn and Fn ¼ Fn for na1 and c1 ¼ c, F1 ¼ F for n ¼ 1. Note that the introduction of the temporal rescaling (30) implies that   d dc ðcn einðO=2Þt Þ ¼ inðO=2Þcn þ q n einðO=2Þt , (35a) dt dt   d inðO=2Þt q dFn ðFn e Þ ¼ inðO=2ÞFn þ  (35b) einðO=2Þt . dt dt In order to determine the coefficients cn ðt; Þ and Fn ðt; Þ, we substitute solution (31) into Eq. (29) and obtain the equations for each harmonic with order n and for a fixed order of approximation on the perturbation parameter e. For n ¼ 0, we obtain O2 Z  c0 þ 2a3 2 jc1 j2 þ 2a4 2 jF1 j2 þ a7 2 ðc1 Fn1 þ cn1 F1 Þ ¼ 0, 4

(36a)

O2 Z  F0 þ b3 2 ðc1 Fn1 þ cn1 F1 Þ þ 2b4 2 jc1 j2 þ 2b5 2 jF1 j2 ¼ 0. 4

(36b)

The correct balance of the terms indicates d ¼ 2. Therefore, we derive the following relation: c0 ¼ 

8a3 jc1 j2 þ 8a4 jF1 j2 þ 4a7 ðFn1 c1 þ cn1 F1 Þ , O2

(37a)

F0 ¼ 

4b3 ðFn1 c1 þ cn1 F1 Þ þ 8b4 jc1 j2 þ 8b5 jF1 j2 . O2

(37b)

For n ¼ 2, taking into account Eq. (35) yields

and the corresponding relations

3 2 4O c2

¼ a3 c21 þ a4 F21  12f 1 þ a7 c1 F1 ,

(38a)

3 2 4O F2

¼ b3 c1 F1 þ b4 c21 þ b5 F21  12f 2 .

(38b)

  4 1 2 2 a3 c1 þ a4 F1  f 1 þ a7 c1 F1 , c2 ¼ 2 3O2

(39a)

  4 1 2 2 f b c F þ b c þ b F  . 3 1 1 4 1 5 1 2 2 3O2

(39b)

F2 ¼

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From the proper balance of the nonlinear and linear terms, for n ¼ 1, we must consider q ¼ 2. Based on Eq. (29), we have       dc O 1 4 40 20 a2  2 a3 f 1 cn1 þ  2 a23 þ 3a6 þ 2 a7 b4 jc1 j2  Oi 1 þ Os1  a1 i c1 þ 2 2 dt 3O 3O 3O     16 16 8 8 16 16 þ  2 a3 a4  2 a4 b3 þ 2a5  2 a7 b5  2 a27 jF1 j2 c1 þ  2 a4 b4 þ 2 a7 b3 O 3O O 3O O O     8 40 20 8 8 þ 2 a3 a7 jc1 j2 F1 þ  2 a4 b5  2 a4 a7 jF1 j2 F1 þ a3 a4  2 a4 b3 þ a5 O 3O 3O 3O2 O    4 8 20 4 4 2 n 2 þ 2 a7 b5 F21 cn1 þ a b  a b a b a4 f 2 Fn1  4 4 3 7 7 3 c1 F1 þ 4a7 c1 F1  O 3O2 3O2 O2 3O2 2 4  2 a7 f 1 Fn1  2 a7 f 2 cn1 ¼ 0, ð40aÞ 3O 3O     dF1 O 1 2 4 b2  2 b3 f 1  2 b5 f 2 Fn1  Oi þ Os2  b1 i F1 þ 2 2 dt 3O 3O      2 2 4 8 16 16 32 n 2  b f þ 2 b4 f 1 c1 þ  2 a3 b3  2 b3  2 b4 b5 þ 2b6 þ 2 a7 b4 c1  F1 2 3 2 3O 3O O 3O O 3O     20 40 2 20 40 þ  2 a4 b3  2 b5 þ 3b7 jF1 j2 F1 þ  2 b3 b4  2 a3 b4 jc1 j2 c1 O 3O 3O 3O   8 16 40 þ  2 a7 b3  2 a4 b4  2 b3 b5 jF1 j2 c1 3O O 3O   4 2 4 8 8 þ  2 b3 þ 2 a3 b3 þ 2 b4 b5 þ 2 b4 a7 c21 Fn1 O 3O 3O O   28 8 4 þ b3 b5 þ 2 a4 b4 þ b6  2 a7 b3 cn1 F21 ¼ 0. 3O2 3O O

ð40bÞ

Based on Eqs. (26) and (28), the differential equation for the evolution of the complex amplitudes c1 and F1 can be derived as dc1 ¼ m1 c1  s1 c1 i þ ða1 þ a2 f 1 þ a11 f 2 Þcn1 i þ ða3 f 2 þ a11 f 1 ÞFn1 þ a4 jc1 j2 c1 i þ a5 jf1 j2 c1 i dt þ a6 jc1 j2 f1 i þ a7 jf1 j2 f1 i þ a8 f21 cn1 i þ a9 c21 fn1 i þ a10 f21 c1 i,

ð41aÞ

dF1 ¼ m2 F1  s2 F1 i þ ðb1 þ b2 f 1 þ b3 f 2 ÞFn1 i þ ðb4 f 2 þ b5 f 1 Þcn1 i þ b6 jc1 j2 F1 i þ b7 jF1 j2 F1 i dt þ b8 jc1 j2 c1 i þ b9 jF1 j2 c1 i þ b10 c21 Fn1 i þ b11 cn1 F21 i.

ð41bÞ

In order to transform Eq. (41) into the Cartesian form, let c1 ¼ x1 þ ix2 ;

F1 ¼ x3 þ ix4 .

(42)

Substituting Eq. (42) into Eq. (41), the averaged equation in the Cartesian form is obtained as follows: dx1 ¼ m1 x1 þ ðs1 þ a1 þ a2 f 1 þ a11 f 2 Þx2 þ ða3 f 2 þ a11 f 1 Þx4 þ a4 x21 x2 þ a4 x32 þ ða10  a8 þ a5 Þx23 x2 dt þ ða5 þ a8  a10 Þx24 x2 þ 2a9 x1 x2 x3 þ ða9 þ a6 Þx21 x4 þ ða6 þ a9 Þx22 x4 þ 2ða8 þ a10 Þx1 x3 x4 þ a7 x34 þ a7 x23 x4 ,

ð43aÞ

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dx2 ¼ ðs1 þ a1 þ a2 f 1 þ a11 f 2 Þx1 þ m1 x2 þ ða3 f 2 þ a11 f 1 Þx3  a4 x31  a4 x22 x1  ða8 þ a5 þ a10 Þx23 x1 dt  ða5  a8  a10 Þx24 x1  2a9 x1 x2 x4  ða9 þ a6 Þx21 x3  ða6  a9 Þx22 x3  2ða8  a10 Þx2 x3 x4  a7 x24 x3  a7 x33 ,

ð43bÞ

dx3 ¼ ðb4 f 2 þ b5 f 1 Þx2 þ m2 x3 þ ðs2 þ b1 þ b2 f 1 þ b3 f 2 Þx4 þ b8 x21 x2 þ ðb6  b10 Þx21 x4 þ ðb6 þ b10 Þx22 x4 dt þ b8 x32  ðb11  b9 Þx23 x2 þ b7 x23 x4 þ ðb9 þ b11 Þx24 x2 þ b7 x34 þ 2b10 x1 x2 x3 þ 2b11 x1 x3 x4 ,

ð43cÞ

dx4 ¼ ðb4 f 2 þ b5 f 1 Þx1 þ ðs2 þ b1 þ b2 f 1 þ b3 f 2 Þx3 þ m2 x4  b8 x31  ðb6 þ b10 Þx21 x3 þ ðb6  b10 Þx22 x3 dt  b8 x22 x1  ðb9 þ b11 Þx23 x1  b7 x33 þ ðb9  b11 Þx24 x1  b7 x24 x3  2b11 x2 x3 x4  2b10 x1 x2 x4 ,

ð43dÞ

where all coefficients can be found in Appendix B. 4. Numerical simulations of periodic and chaotic motions In the following investigation, the fourth-order Runge–Kutta algorithm [36] is utilized to numerically analyze the periodic and chaotic motions of the FGM rectangular plate subjected to thermal and mechanical loads for the case of 1:1 internal resonance and primary parametric resonance. We consider the averaged equation (43) to carry out numerical simulation. We choose the forcing excitations f1 and f2 as the controlling parameters when the periodic and chaotic responses of the FGM rectangular plate are investigated. Zirconia and titanium alloy are selected for the two constituent materials of the FGM plate in this example, referred to as ZrO2/Ti–6Al–4V. The properties of this material can be found in Ref. [37]. The two-dimensional phase portrait, waveform, three-dimensional phase portrait and Poincare map are plotted to demonstrate the nonlinear dynamic behaviors of the FGM rectangular plate. It can be clearly found from the numerical results that the periodic and chaotic motions occur for the FGM rectangular plate. Fig. 2 illustrates the existence of the chaotic motion for the FGM rectangular plate when the forcing excitation f2 is 6.99. The parameters and the initial conditions are, respectively, chosen as s1 ¼ 0:52, s2 ¼ 0:86, m1 ¼ 0:32, m2 ¼ 0:32, a1 ¼ 5:2, a2 ¼ 12:2, a3 ¼ 8:1,a4 ¼ 0:66, a5 ¼ 0:5, a6 ¼ 4:21, a7 ¼ 0:36, a8 ¼ 3:3, a9 ¼ 12:2, a10 ¼ 3:26, a11 ¼ 5:78, b1 ¼ 4:4, b2 ¼ 15:2, b3 ¼ 8:73, b4 ¼ 3:03, b5 ¼ 5:1, b6 ¼ 2:2, b7 ¼ 6:98, b8 ¼ 11:3, b9 ¼ 5:21, b10 ¼ 4:15, b11 ¼ 0:66, f 1 ¼ 8:62, x10 ¼ 0:44, x20 ¼ 1:55, x30 ¼ 2:35, x40 ¼ 5:18. Figs. 2(a) and (c) represent the phase portraits on the planes (x1,x2) and (x3,x4), respectively. Figs. 2(b) and (d) respectively denote the waveforms on the planes (t,x1) and (t,x3). Figs. 2(e) and (f) represent the three-dimensional phase portrait in space (x1, x2, x3) and the Poincare map on plane (x1, x2), respectively. It can be shown from Fig. 2 that the amplitude of the second-order mode is larger than one of the first-order mode. Until the forcing excitation is increased to f2 ¼ 7.008, the response of the FGM rectangular plate also is the chaotic motion, as shown in Fig. 3. The Poincare maps given in Figs. 2(f)–3(f) clearly demonstrate that chaotic motions exist for the FGM rectangular plate. Fig. 4 indicates that the quasi-period motion of the FGM rectangular plate occurs when the forcing excitation changes to f2 ¼ 7.012. Fig. 5 shows that the chaotic motions of the FGM rectangular plate again occur when the forcing excitation changes to f2 ¼ 7.038. Fig. 6 illustrates that the quasi-period response of the FGM rectangular plate occurs when f2 ¼ 7.053. When f2 ¼ 7.098, the chaotic response of the FGM rectangular plate exists for the FGM rectangular plate, as shown in Fig. 7. Continuously increasing the forcing excitation to f2 ¼ 7.288, it is found that period-9 solution occurs for the FGM rectangular plate, as shown in Fig. 8. In Fig. 9, it is seen that the multiple period motion of the FGM rectangular plate exists when f2 ¼ 7.389. With the increasing of the forcing excitation continuously, the periodic motion of the FGM rectangular plate also exists when f2 ¼ 7.403, as shown in Fig. 10. Because of limited space, we do not provide other figures.

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Fig. 2. The chaotic motion of the FGMs rectangular plate exists when f2 ¼ 6.99, (a) the phase portrait on plane (x1, x2); (b) the waveforms on the planes (t, x1); (c) the phase portrait on plane (x3, x4); (d) the waveforms on the planes (t, x3); (e) three-dimensional phase portrait in space (x1, x2, x3); and (f) the Poincare map on plane (x1, x2).

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Fig. 3. The chaotic motion of the FGMs rectangular plate exists when f2 ¼ 7.008.

From Figs. 2–10, it can be shown that the process of change for the motions of the FGM rectangular plate is as follows: the chaotic motion-the quasi-period motion-the chaotic motion-the quasi-period motionthe chaotic motion-the period n motion.

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Fig. 4. The quasi-period motion of the FGMs rectangular plate exists when f2 ¼ 7.012.

877

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Fig. 5. The chaotic motion of the FGMs rectangular plate exists when f2 ¼ 7.038.

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Fig. 6. The quasi-period motion of the FGMs rectangular plate exists when f2 ¼ 7.053.

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Fig. 7. The chaotic motion of the FGMs rectangular plate exists when f2 ¼ 7.098.

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Fig. 8. The period-9 motion of the FGMs rectangular plate exists when f2 ¼ 7.288.

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Fig. 9. The multiple period motion of the FGMs rectangular plate exists when f2 ¼ 7.389.

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Fig. 10. The period motion of the FGMs rectangular plate exists when f2 ¼ 7.403.

5. Conclusions The nonlinear oscillations and chaotic dynamics of the FGM rectangular plate under the combined transverse and in-plane excitations are investigated for the first time. The material properties are assumed to

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be temperature dependent. Based on Reddy’s third-order plate theory, the governing equations of motion for the FGM rectangular plate are derived using the Hamilton’s principle. Only transverse nonlinear oscillations of the FGM plate are considered. Galerkin’s approach is utilized to discretize the governing equation of motion to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:1 internal resonance and principal parametric resonance-1/2 subharmonic resonance. The asymptotic perturbation method based on the Fourier expansion and the temporal rescaling is utilized to obtain a four-dimensional nonlinear averaged equation. Using the fourth-order Runge–Kutta algorithm, the averaged equation is analyzed numerically. Under certain conditions, the periodic, quasiperiodic and chaotic motions of the FGM rectangular plate are found. The influence of the forcing excitations f1 and f2 on the nonlinear dynamic behaviors of the FGM rectangular plate is investigated. It is thought that the forcing excitations f1 and f2 can change the form of motions for the FGM rectangular plate. In the situation investigated in this paper, the forcing excitation can be considered to be a controlling force, which can control the responses of the FGM rectangular plate from the chaotic motion to the period n or quasi-period motions. Because the main interest in this paper is focused on the analytical and numerical researches on the nonlinear oscillations and chaotic dynamics of the FGM rectangular plate under combined transverse and inplane excitations, we did not find similar work conducted to date. Thus, it is a difficult job for us to give some comparisons with the jobs of other researchers. In this paper, we mainly give the qualitative analysis on nonlinear dynamics of the FGMs rectangular plate rather than the quantitative analysis. Therefore, we did not clearly give the temperature rise DT or surface temperature T1 and T2. It is thought that the external loads are the fast varying excitations and the temperature rise DT or surface temperatures T1 and T2 are the slow varying excitations. It is a new problem on how to study the influence of both the fast and slow varying excitations on nonlinear responses of the FGM plate.

Acknowledgments The authors gratefully acknowledge the support of the National Science Foundation for Distinguished Young Scholars of China (NSFDYSC) through Grant no. 10425209, the National Natural Science Foundation of China (NNSFC) through Grant no. 10732020, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHRIHLB) and the China Postdoctoral Science Foundation through Grant no. 20060390386.

Appendix A The coefficients ai (i ¼ 1, y, 6) and bj (j ¼ 1, y, 7) presented in Eq. (26) are as follows: m007 þ p0 m008 m004 m008 p1 m009 m010 m011 ; a1 ¼  ; a2 ¼  ; a3 ¼  ; a4 ¼  ; a5 ¼  , m001 m001 m001 m001 m001 m001 m012 m014 m013 n007 þ P0 n008 n005 n008 P1 a6 ¼  ; a7 ¼  ; f1 ¼ F 1 ; o22 ¼  ; b1 ¼  ; b2 ¼ , m001 m001 m001 n002 n002 n002 n006 n009 n010 n011 n012 n013 ; b4 ¼  ; b5 ¼  ; b6 ¼  ; b7 ¼  ; f2 ¼ F 2, ðA:1Þ b3 ¼  n002 n002 n002 n002 n002 n002

o21 ¼ 

where 1 1 1 m001 ¼  3pc1 ðI 4  c1 I 6 Þðam09 þ bl09 Þ=4  I 0 ab; m004 ¼ gab; m008 ¼  bp2 , 4 4 4a  1 bp2 4ap2 b 81a  m007 ¼ ðA55 þ c22 F 55  2c2 D55 Þ  ðA44 þ c22 F 44  2c2 D44 Þ 3 c21 H 11 p4  3 c21 H 22 p4 4 a a 9b 4b 

9p4 2 1 b ðc H 21 þ 2c21 H 66 Þ  pbm09 ðA55 þ c22 F 55  2c2 D55 Þ þ 2 p3 m09 c1 ðF 11  c1 H 11 Þ 4 a 2ab 1

ARTICLE IN PRESS Y.X. Hao et al. / Journal of Sound and Vibration 312 (2008) 862–892

3 3 3 p l09 c1 ðF 12 þ 2F 66  c1 H 21  2c1 H 66 Þ  pal09 ðA44 þ c22 F 44  2c2 D44 Þ 4a 4  a 3 þ 2 p l09 c1 ðF 22  c1 H 22 Þ , b

þ

m009 ¼

m010 ¼

32bp 32p 32ap c1 ðE 21 þ 2E 66 Þð175l03 Þ þ c1 E 11 ð175l03 þ 525l07 Þ þ c1 E 22 ð175m03 Þ 4275a2 525b 175b2 32p 4096p2 c1 16p c1 ðE 21 þ 2E 66 Þð175m03 Þ  m ðB21  c1 E 21 Þ þ ð2E 66  E 12 Þ þ 1575a 3b 09 3ab 8p 8p 56bp m ðB11  c1 E 11 Þ  m09 ðB21 þ B66  c1 E 21  c1 E 66 Þ  m09 ðB66  c1 E 66 Þ þ 3b 3b 171a2 09 16a 4p 16 8a þ 2 pl09 ðB22  c1 E 22 Þ þ l09 ðB66  c1 E 66 Þ þ pl09 ðB12  c1 E 12 Þ  2 pl09 ðB22  c1 E 22 Þ 9a 9a b 9b 8p 8p  l09 ðB66  c1 E 66 Þ  l09 ðB21 þ B66  c1 E 21  c1 E 66 Þ, 9a 9a 32bp 864p 96ap c1 l06 ðE 21 þ 2E 66 Þ þ c1 E 11 ð2178l06 þ 1215l08 Þ þ c1 E 22 m06 4275a2 525b 175b2 7776p 4156c1 2 432p c1 m06 ðE 21 þ 2E 66 Þ þ m ðB21  c1 E 21 Þ p ð2E 66  E 12 Þ þ þ 1575a 525b 10 525ab 1512p 21384b 408p m ðB21 þ B66  c1 E 21  c1 E 66 Þ þ m ðB66  c1 E 66 Þ pm ðB11  c1 E 11 Þ þ  525b 10 4275a2 10 525b 10 48ap 12852 3888p pl10 ðB66  c1 E 66 Þ  l10 ðB12  c1 E 12 Þ  l ðB22  c1 E 22 Þ þ 2 10 1575a 1575a 175b 168a 13608 3672p pl10 ðB66  c1 E 66 Þ þ l10 ðB21 þ B66  c1 E 21  c1 E 66 Þ,  pl10 ðB22  c1 E 22 Þ þ 2 1575a 1575a 157b

m011 ¼

m012 ¼

p3 bp3 A12 ðl01 þ l02  2l05 þ 2l04 Þ  2 A11 ð9l01 þ 9l02  18l05 þ 18l04 Þ 8b 8a bp3 3p3  2 A11 ð12l05 þ 3l02  3l01 þ 12l04 Þ  A66 ðl01  2l02 þ 2l05 þ l04 Þ 4a 8b p3 3p3 A66 ð4l05 þ 4l02  l01 þ l04 Þ þ ðA21 þ A66 Þð4l05  2l02  l01 þ 2l04 Þ  4b 4b 3p3 p3 þ A66 ð2m04 þ m02 þ m01 þ 2m05 Þ  A22 ðm04  2m02 þ m01 þ 2m05 Þ 4a 8a 3p3 p3 a A12 ð3m04  6m02 þ 3m01 6m05 Þ  2 A22 ðm04 þ 4m02 þ m01  4m05 Þ  8a 4b 3 p 3p3  A66 ð4m04 þ m02 þ m01  4m05 Þ þ ð2m04  2m02 þ m01 þ 4m05 ÞðA66 þ A12 Þ 4a 4a     27p4 a 5p4 1 27p4 4bp3 bp4 9p3 9p4 A21 þ A66   l08 þ A22 þ þ 2 l08 þ 3 þ A11 , ab 2 32a3 a a b ab 32b3

p3 bp3 bp3 9p3 ð9l03 þ 18l07 ÞA12 þ 2 ðl03 þ 2l07 ÞA11  2 ðl03 þ 2l07 ÞA11  l03 A66 8b 8a 4a 8b 9p3 9p3 3p3 27p3 3p3 27p3 a ðA21 þ A66 Þl03 þ l03 A66  m03 A66  m03 A22  m03 A12 þ þ m03 A22 4b 4b 4a 8a 8a 4b2

885

ARTICLE IN PRESS Y.X. Hao et al. / Journal of Sound and Vibration 312 (2008) 862–892

886

  3p3 3p3 729p4 a 27p4 1 9bp4 A m03 A66 þ m03 ðA66 þ A12 Þ  þ A  þ A A11  22 21 66 4a 4a 32ab 2 128a3 128b3  3  bp bp4 9p3 9p4 l  l þ þ þ A11 , 07 07 4a2 16a3 4b 16ab

m014 ¼

32bp c1 E 11 ð675l02 þ 1512l04  1080l05 þ 945l10 Þ 4275a2 32p c1 ðE 21 þ 2E 66 Þð42l04 þ 105m01  1200m05  300l02 Þ þ 525b 32ap þ c1 E 22 ð7m04 þ 35m01  40m05  200m02 Þ 175b2 32pc1 ðE 21 þ 2E 66 Þð252m04 þ 315m01  360m05  450m02 Þ þ 1575a 24912c1 2 16p ðB21  c1 E 21 Þð9m09 þ 243m10 Þ p ð2E 66  E 12 Þ   525b 525ab 8p ðB21 þ B66  c1 E 21  c1 E 66 Þð9m09 þ 27m10 Þ þ 525b 8bp 8p ðB66  c1 E 66 Þð729m09 þ 27m10 Þ ðB11  c1 E 11 Þð729m09  2687m10 Þ þ þ 2 4275a 525b 16ap 4p ðB66  c1 E 66 Þð243l10 þ 81l09 Þ  ðB22  c1 E 22 Þð27l10 þ 9l09 Þ  2 1575a 175b 16p 8ap ðB12  c1 E 12 Þð27l10 þ 729l09 Þ þ  ðB22  c1 E 22 Þð3l10 þ 9l09 Þ 1575a 1575b2 8p 8p ðB66  c1 E 66 Þð243l10 þ 9l09 Þ þ ðB21 þ B66  c1 E 21  c1 E 66 Þð279l09 þ 243l10 Þ,  1575a 1575a ab , 4 ¼  3pc1 ðI 4  c1 I 6 Þðam09 þ bl09 Þ=4  14I 0 ab;

m13 ¼ n002

n005 ¼ 14gab,

32bp 32p c1 ðE 21 þ 2E 66 Þð252l02 c1 E 11 ð35l01  7l02 þ 40l05 þ 200l04 Þ  2 175a 1575b 32ap  315l01 þ 450l04 þ 360l05 Þ þ c1 E 22 ð945m01  675m04  1080m05 þ 1512m02 Þ 4725b2 32p 23636p2 c1 ðE 12 þ 2E 66 Þð105m01 þ 42m02  120m03  300m04 Þ  þ c1 ðE 66  E 12 Þ 525a 525ab 1269p2 16p ðB21  c1 E 21 Þð27m09 þ 729m10 Þ  c1 ðE 12  2E 66 Þ  1575b 525ab 8p 8bp ðB21 þ B66  c1 E 21  c1 E 66 Þð243m09 þ 729m10 Þ þ ðB11  c1 E 11 Þð3m09 þ 9m10 Þ þ 1575b 175a2 16bp 8p ðB66  c1 E 66 Þð243m09 þ 9m10 Þ ðB11  c1 E 11 Þð27m09 þ 9m10 Þ   2 175a 1575b 16pa 8p ðB66  c1 E 66 Þð27l09 þ 81l10 Þ  ðB22  c1 E 22 Þð243l09 þ 729l10 Þ  2 525a 4725b 16p 8ap ðB12  c1 E 12 Þð243l09 þ 9l10 Þ þ þ ðB22  c1 E 22 Þð2187l09  729l10 Þ 525a 4725b2 8p 8p ðB66  c1 E 66 Þð27l09 þ 279l10 Þ þ ðB12 þ B66  c1 E 12  c1 E 66 Þð27l09 þ 9l10 Þ, þ 525a 525a

n006 ¼ 

ARTICLE IN PRESS Y.X. Hao et al. / Journal of Sound and Vibration 312 (2008) 862–892

9bp2 ap2 81bp4 2 ap4 2 ðA55 þ c22 F 55  2c2 D55 Þ  ðA44 þ c22 F 44  2c2 D44 Þ  c H  c1 H 22 11 4a 4b 4a3 1 4b3 9p4 2 3 27b ðc1 H 21 þ 2c21 H 66 Þ  pbm10 ðA55 þ c22 F 55  2c2 D55 Þ þ 2 p3 m10 c1 ðF 11  c21 H 11 Þ  4 4a 2ab 1 9 3 2  pal10 ðA44 þ c2 F 44  2c2 D44 Þ þ p l10 c1 ðF 12 þ 2F 66  c1 H 21  2c1 H 66 Þ 4 4a a þ 3 p4 l10 c1 ðF 22  c1 H 22 Þ, 4b 9 1 ab ¼  bp2  ap2 ; n13 ¼ , 4a 4b 4

n007 ¼ 

n008

32bp 7776p 69984ap c1 l03 ðE 21 þ 2E 66 Þ þ c1 E 11 ð105l07  3l03 Þ þ c1 E 22 m03 2 175a 1575b 4725b2 864p 1296p2 4284p2 c1 m03 ðE 12 þ 2E 66 Þ  c1 ðE 66  E 12 Þ  c1 ðE 12  2E 66 Þ þ 525a 525ab 525ab 3888p 3672p m09 ðB21  c1 E 21 Þ þ m ðB21 þ B66  c1 E 21  c1 E 66 Þ  1575b 1575b 09 8bp 48bp 13608p m ðB66  c1 E 66 Þ m09 ðB11  c1 E 11 Þ   2 m10 ðB11  c1 E 11 Þ  2 a 175a 1575b 09 34992pa 8p 432p l09 ðB12  c1 E 12 Þ  l09 ðB22  c1 E 22 Þ þ l10 ðB66  c1 E 66 Þ  3a 525a 4725b2 33048ap 408p 1512p l09 ðB66  c1 E 66 Þ þ l09 ðB12 þ B66  c1 E 12  c1 E 66 Þ, þ l ðB22  c1 E 22 Þ þ 2 09 525a 525a 4725b

n009 ¼ 

32bp 224p 32ap c1 l06 ðE 21 þ 2E 66 Þ  c1 E 11 ð175l06  525l08 Þ  c1 E 22 m06 2 175a 63b 27b2 32p 8p2 42p2 m06 c1 ðE 12 þ 2E 66 Þ  c1 ð2E 66  E 12 Þ  c1 ðE 12  2E 66 Þ  3a 3ab ab 16p 8p 168bp m10 ðB21  c1 E 21 Þ  m10 ðB21 þ B66  c1 E 21  c1 E 66 Þ  m ðB11  c1 E 11 Þ þ 9b 9b 175a2 09 16bp 14008p 9p4 16pa m09 þ ðA21 þ 2A66 Þ þ l10 ðB22  c1 E 22 Þ þ 2 m09 ðB11  c1 E 11 Þ  ðB66  c1 E 66 Þ a 1575b 23ab 27b2 120p 16p 8ap 8p l09 ðB66  c1 E 66 Þ þ l10 ðB12  c1 E 12 Þ  þ l10 ðB22  c1 E 22 Þ  l09 ðB66  c1 E 66 Þ 2 21a 3a 3a 27b 8p  l10 ðB12 þ B66  c1 E 12  c1 E 66 Þ, 3a

n010 ¼ 

3 3 b p A21 ð6l04 þ 3l01  3l02 þ 6l05 Þ  2 p3 A11 ð2l05  2l04 þ l01  l02 Þ 8b 8a 3 3 þ p3 A66 ð2l05  l04  l01 þ 2l02 Þ þ p3 ð4l05  2l04 þ l01  2l02 ÞðA66 þ A21 Þ 4b 4b b 1 þ 2 p3 A11 ð4l05  4l04  l01 þ l02 Þ þ p3 A66 ð4l05  l04  l01 þ 4l02 Þ 4a 4b 3 1 þ p3 A66 ð2m01 þ 2m04  m02  2m05 Þ  p3 A12 ðm01  m04  2m02 þ 2m05 Þ 4a 8a 3 1 þ p3 A66 ð2m01 þ 2m04  m02  2m05 Þ  p3 ðm01 þ 2m04  2m02  4m05 ÞðA66 þ A12 Þ 4a 4a

n011 ¼ 

887

ARTICLE IN PRESS Y.X. Hao et al. / Journal of Sound and Vibration 312 (2008) 862–892

888

    27ap4 p4 1 81p4 1 729bp4 A A  A  þ A þ A A11   22 12 66 12 66 16ab 2 16ab 2 128a3 32b3   9b 3 9b 4 1 3 1 4 p l07 þ p A11  p l07 þ p þ 2 3 4a 16a 4ab 16ab 

ap3 ap3 A ð9m  18m þ 18m  9m Þ þ A22 ð3m01  12m02 þ 12m05  3m04 Þ, 22 01 02 05 04 8b2 4b2

3 3 b 3 3 p ð6l06  2l08 ÞA21  2 p3 ð27l06  54l08 ÞA11  p3 l06 A66 þ p3 l06 ðA66 þ A21 Þ 8b 8a 4b 4b b 3 3 3 9 3 a 3 9  2 p ð54l08  27l06 ÞA11 þ p l06 A66  p m06 A66 þ 2 p m06 A22  p3 m06 A12 4a 4b 4a 8a 8b     9 3 9ap4 27 4 1 27 4 1 p A12 þ A66  p A12 þ A66 þ p m06 ðA66 þ A12 Þ  A22  4a 64ab 2 64ab 2 128b3   27bp4 9b 3 9b 4 1 3 1 4 p p  A  p l þ p þ l þ A11 , 11 08 08 a2 4a3 ab 4ab 32a3

n012 ¼ 

ðA:2Þ

where gi2 l i1  gi1 l i2 ki2 l i1  ki1 l i2 ; li ¼ ði ¼ 1; . . . ; 10Þ, gi2 ki1  gi1 ki2 ki2 gi1  ki1 gi2 4 4 4 4 X X X X gi1 ¼ ðx1j Þ2 ; gi2 ¼ x1j z1j ; ki2 ¼ ðz1j Þ2 ; ki1 ¼ x1j z1j , mi ¼

l i1 ¼

j¼1

j¼1

4 X

4 X

x1j Z1j ;

l i2 ¼

j¼1

 1 1 q þ q ; a2 11 b2 12   1 4 ¼ 4p2 2 q11 þ 2 q12 ; a b   1 9 ¼ 4p2 2 q11 þ 2 q12 ; a b   1 2 4 ¼ 4p q þ q ; a2 11 b2 12   1 2 1 ¼ 16p q þ q ; a2 11 b2 12

x11 ¼ 4p2 x21 x31 x41 x51



x71 x83 x93 x102

j¼1

ði ¼ 1; . . . ; 10Þ,

z1j Z1j

j¼1

 1 1 p2 q þ q x14 ¼ 4 q43 , 31 32 ; 2 2 a ab b   2 8p 1 4 8p2 q23 ; x23 ¼ 4p2 2 q31 þ 2 q32 ; x24 ¼ q , x22 ¼ a ab ab 43 b   12p2 1 12p2 q23 ; x33 ¼ 4p2 2 q31 þ 9q32 ; x34 ¼ q , x32 ¼ a ab ab 43   8p2 1 8p2 2 4 q23 ; x43 ¼ 4p q , q þ q ¼ x42 ¼ ; x 44 31 32 a2 ab ab 43 b2   16p2 1 1 16p2 q23 ; x53 ¼ 16p2 2 q31 þ 2 q32 ; x54 ¼ q , x52 ¼ a ab ab 43 b x12 ¼

4p2 q ; ab 23



x13 ¼ 4p2



 9 1 12p2 q , q þ q x64 ¼ 31 32 ; 2 2 a ab 43 b 4p2 4p2 36p2 ¼ 2 q11 ; x72 ¼ 0; x73 ¼ 2 q31 ; x74 ¼ 0; x81 ¼ 2 q11 ; x82 ¼ 0, a a a   36p2 1 9 3p2 q , ¼ 2 q31 ; x84 ¼ 0; x91 ¼ p2 2 q14 þ 2 q15 ; x92 ¼ a a ab 26 b     1 9 3p2 9 1 q46 ; x101 ¼ p2 2 q14 þ 2 q15 , ¼ p2 2 q34 þ 2 q35 þ q37 ; x94 ¼ a a ab b b   2 2 3p 9 1 3p 4p2 q26 ; x103 ¼ p2 2 q34 þ 2 q35 þ q37 ; x104 ¼ q46 ; z11 ¼ q , ¼ a ab ab ab 13 b

x61 ¼ 4p2

 9 1 q þ q ; a2 11 b2 12

j¼1

x62 ¼

12p2 q ; ab 23

x63 ¼ 4p2



ðA:3Þ

ARTICLE IN PRESS Y.X. Hao et al. / Journal of Sound and Vibration 312 (2008) 862–892

z12 z22 z32 z42 z52

 1 1 4p2 q ; ¼ 4p q þ q ¼ ; z 13 a2 21 b2 22 ab 33   1 4 8p2 q ; ¼ 4p2 2 q21 þ 2 q22 ; z23 ¼ a ab 33 b   1 9 12p2 q ; ¼ 4p2 2 q21 þ 2 q22 ; z33 ¼ a ab 33 b   4 1 8p2 q ; ¼ 4p2 2 q21 þ 2 q22 ; z43 ¼ a ab 33 b   16 16p2 2 16 q ; ¼p q þ q ¼ ; z 53 21 22 2 a2 ab 33 b 2



z62 ¼ 4p

2



 9 1 q þ q ; a2 21 b2 22

z63

 1 1 z14 ¼ 4p q þ q ; a2 41 b2 42   1 4 z24 ¼ 4p2 2 q41 þ 2 q42 ; a b   1 9 z34 ¼ 4p2 2 q41 þ 2 q42 ; a b   4 1 z44 ¼ 4p2 2 q41 þ 2 q42 ; a b   1 2 1 z54 ¼ 16p q þ q ; a2 41 b2 42

12p2 q ; ¼ ab 33

2

889



z64 ¼ 4p

2

z21 ¼

8p2 q , ab 13

z31 ¼

12p2 q , ab 13

z41 ¼

8p2 q , ab 13

12p2 q , ab 13 12p2 q , ¼ ab 13

z51 ¼

 9 1 q þ q ; a2 41 b2 42

z61



z71 ¼ 0,

4p2 4p2 36p2 q ; z73 ¼ 0; z74 ¼ 2 q42 ; z81 ¼ 0; z82 ¼ 2 q22 ; z83 ¼ 0, 2 22 b b b   36p2 3p2 1 9 3p2 q16 ; z92 ¼ p2 2 q24 þ 2 q25 ; z93 ¼ q , ¼ 2 q42 ; z91 ¼ a ab ab 36 b b     1 9 3p2 9 1 q16 ; z102 ¼ 4p2 2 q24 þ 2 q25 , ¼ p2 2 q44 þ 2 q45 þ q47 ; z101 ¼ a a ab b b     2 3 3p 9 1 p 3 14 3 q36 ; z104 ¼ p2 2 q44 þ 2 q45  q47 ; Z11 ¼ ¼ s þ s  s 11 12 13 , a ab 2a a2 b b2 b2

z72 ¼ z84 z94 z103

Z12 ¼ Z14 ¼ Z22 ¼ Z24 ¼ Z32 ¼

Z34 Z42 Z44 Z52 Z54

    p3 14 3 3 p3 3 14 3 s21  2 s22 þ 2 s23 ; Z13 ¼ s31 þ 2 s32 þ 2 s33 , a 2b a2 2a a2 b b b     p3 14 3 3 p3 3 13 6 s41 þ 2 s42 þ 2 s43 ; Z21 ¼  s11 þ 2 s12 þ 2 s13 , a 2b a2 2a a2 b b b     p3 7 3 3 p3 3 13 6 s21  2 s22 þ 2 s23 ; Z23 ¼ s31 þ 2 s32 þ 2 s33 , a b a2 2a a2 b b b     p3 2 3 3 p3 1 9 3 s  s þ s ¼ s þ s þ s ; Z , 41 42 43 11 12 13 31 a2 b a2 4a a2 b2 b2 b2     p3 3 27 9 p3 1 9 3  s þ s þ s ¼ s þ s þ s ; Z 21 22 23 31 32 33 , 33 a2 4b a2 a a2 b2 b2 b2

    p3 3 27 9 p3 3 7 3 ¼ s41 þ 2 s42 þ 2 s43 ; Z41 ¼  s11 þ 2 s12 þ 2 s13 , a 4b a2 a a2 b b 2b     p3 13 6 3 p3 3 7 3  2 s21 þ 2 s22 þ 2 s23 ; Z43 ¼  ¼ s31 þ 2 s32 þ 2 s33 , a a 4b a a2 b b 2b     p3 13 3 3 p3 3 7 3  2 s41 þ 2 s42 þ 2 s43 ; Z51 ¼ ¼ s11 þ 2 s12 þ 2 s13 , a a 2b a a2 b b b     p3 7 3 3 p3 3 7 3 ¼ s þ s þ s ¼ s þ s þ s ; Z , 21 22 23 31 32 33 53 a2 b a2 a a2 b2 b2 b2     p3 7 3 3 p3 27 3 9 ¼ s þ s þ s ¼ s þ s þ s ; Z 41 42 43 11 12 13 , 61 a2 b a2 4a a2 b2 b2 b2

ARTICLE IN PRESS Y.X. Hao et al. / Journal of Sound and Vibration 312 (2008) 862–892

890

Z62 Z64 Z72 Z74 Z82

    p3 9 1 3 p3 27 3 9 ¼ s21 þ 2 s22 þ 2 s23 ; Z63 ¼ s31 þ 2 s32 þ 2 s33 , a 4b a2 4a a2 b b b     p3 9 1 3 p3 1 9 3  ¼ s þ s þ s ¼ s  s þ s ; Z , 41 42 43 11 12 13 71 a2 a2 4b a2 4a b2 b2 b2     p3 3 27 9 p3 1 9 3  ¼ s  s þ s ¼ s  s þ s ; Z 21 22 23 31 32 33 , 73 a2 a2 4b a2 4a b2 b2 b2     p3 3 27 9 p3 27 3 9  2 s41  2 s42 þ 2 s43 ; Z81 ¼  2 s11  2 s12 þ 2 s13 , ¼ a a a 4b 4a b b b     3 3 p 9 1 3 p 27 3 9  2 s21  2 s22 þ 2 s23 ; Z83 ¼  2 s31  2 s32 þ 2 s33 , ¼ a a a 4b 4a b b b     p3 9 1 3 p3 1 9  2 s41  2 s42 þ 2 s43 ; Z91 ¼  Z84 ¼ s14 þ 2 s15 , a a 4b a a2 b b     p3 27 3 p p2 9p2 Z92 ¼  s24 þ 2 s25 ; Z93 ¼  s34 þ 2 s35  s36 , a a a2 b b2 b     p p2 9p2 p3 27 3 Z94 ¼  s þ s  s ¼ s þ s ; Z , 44 45 46 14 15 101 b b2 a2 a a2 b2     p3 1 9 p 27p2 3p2 Z102 ¼ s þ s ¼ s þ s þ 3s ; Z 24 25 34 35 36 , 103 a2 a a2 b b2 b2   p p2 9p2 Z104 ¼ s þ s þ s 44 45 46 , b b2 a2

ðA:4Þ

and q11 ¼ s11 ¼ A11 ; q12 ¼ q21 ¼ s12 ¼ s21 ¼ A66 ; q13 ¼ q23 ¼ s13 ¼ s23 ¼ A12 þ A66 , q14 ¼ q25 ¼ q31 ¼ q42 ¼ s31 ¼ s42 ¼ B11  c1 E 11 ; s15 ¼ s25 ¼ c1 E 12  2c1 E 66 , q15 ¼ q24 ¼ q32 ¼ q42 ¼ s32 ¼ s41 ¼ B66  c1 E 66 ;

q22 ¼ s22 ¼ A22 ;

s14 ¼ s24 ¼ c1 E 11 ,

q16 ¼ q26 ¼ q32 ¼ q33 ¼ q43 ¼ s33 ¼ s43 ¼ B12  c1 E 12 þ B66  c1 E 66 , q34 ¼ q45 ¼ D11  2c1 F 11 þ c21 H 11 ;

q35 ¼ q44 ¼ D66  2c1 F 66 þ c21 H 66 ,

q36 ¼ q47 ¼ s36 ¼ s46 ¼ ðA55  2c1 D55 þ c22 F 55 Þ;

s34 ¼ s44 ¼ c1 F 11 þ c21 H 11 ,

s35 ¼ s45 ¼ c1 F 12  2c1 F 66 þ c21 H 12 þ 2c21 H 66 , q37 ¼ q46 ¼ D12  2c1 F 12  2c1 F 66 þ D66 þ c21 H 12 þ c21 H 66 . Appendix B The coefficients given in Eq. (43) are as follows: 1 1 4 4 40 3 a2 ; a2 ¼  a3 ; a3 ¼  a4 ; a4 ¼  3 a23  a6 , m1 ¼  a1 ; a1 ¼  2 2O 3O 3O O 3O 16 16 2 8 8 2 20 40 a5 ¼  3 a3 a4 þ 3 a4 b3 þ a5  3 a7 b5  3 a7 ; a7 ¼  3 a7 a4  3 a4 b5 , O O 3O O 3O 3O 3O 16 8 16 20 8 4 a6 ¼ 3 a4 b4 þ 3 a3 a7  3 a7 b3 ; a 9 ¼  3 a3 a4 þ 3 a4 b4  3 a7 b3 , O 3O 3O 3O 3O O 8 4 8 1 4 2 2 1 a7 ; m2 ¼  b1 , a8 ¼ a a þ 3 a7 b5 þ 3 a4 b3 þ a5 ; a10 ¼ a7 ; a11 ¼  3 3 4 O O 3O 2 3O 3O O 1 2 4 2 4 b 2 ; b2 ¼  3 b 3 ; b 3 ¼  3 b 5 ; b4 ¼ b1 ¼ b 3 ; b5 ¼ b4 , 2O 3O 3O 3O3 3O3

ðA:5Þ

ARTICLE IN PRESS Y.X. Hao et al. / Journal of Sound and Vibration 312 (2008) 862–892

8 16 16 2 32 20 40 3 a b þ 3 b23  3 b4 b5  b6 þ 3 a7 b4 ; b7 ¼  3 a4 b3  3 b25 þ b7 , 3 3 3 O O O 3O O 3O O 3O 20 40 8 16 40 b8 ¼  3 b4 b3  3 a3 b4 ; b9 ¼  3 b4 b3  3 a4 b4  3 b3 b5 , 3O 3O 3O O O 4 2 4 8 8 28 8 b10 ¼  3 b3  3 a3 b3  3 b4 b5  a7 b4 ; b11 ¼  3 b3 b5  3 a4 b4 . O O 3O 3O 3O 3O

891

b6 ¼ 

ðB:1Þ

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