Nonlinear dynamic response of a simply supported rectangular functionally graded material plate under the time-dependent thermalmechanical loads

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1 Journal of Mechanical Science and echnolog 5 (7 (11 167~ DOI 1.17/s Nonlinear dnamic response of a simpl supported rectangular functionall graded material plate under the time-dependent thermalmechanical loads Y. X. Hao 1 W. Zhang * J. Yang and S. Y. Li 4 1 College of Mechanical Engineering Beijing Information Science and echnolog Universit Beijing 119 China College of Mechanical Engineering Beijing Universit of echnolog Beijing 14 China School of Aerospace Mechanical and Manufacturing Engineering RMI Universit PO Bo 71 Bundoora VIC 8 Australia 4 ianjin ke labrator of Information Sensing and Intelligent Control tianjin Universit of echnolog and Education ianjin China (Manuscript Received April 1; Revised Jul 1; Accepted Februar Abstract An analsis on nonlinear dnamic characteristics of a simpl supported functionall graded materials (FGMs rectangular plate subjected to the transversal and in-plane ecitations is presented in the time dependent thermal environment. Here we look the FGM Plates as isotropic materials which is assumed to be temperature dependent and graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. he geometrical nonlinearit using Von Karman s assumption is introduced. he formulation also includes in-plane and rotar inertia effects. In the framework of Redd s third-order shear deformation plate theor the governing equations of motion for the FGM plate are derived b the Hamilton s principle. hen the equations of motion with twodegree-of-freedom under combined the time-dependent thermomechanical loads can be obtained b using Galerkin s method. Using numerical method the control equations are analzed to obtain the response curves. Under certain conditions the periodic and chaotic motions of the FGM plate are found. It is found that because of the eistence of the temperature which relate to the time the motions of the FGM plate show the great difference. A period motion can be changed into the chaotic motions which are affected b the time dependent temperature. Kewords: Functionall graded materials; Rectangular plates; Chaotic motion; Higher-order theor; he time-dependent thermalmechanical loads Introduction Functionall graded material (FGM used initiall as thermal barrier materials for aerospace structural applications and fusion reactors are now developed for the general use as structural components in high temperature environments and being strongl considered as a potential structural material candidate for the design of high speed aerospace vehicles [1]. FGMs are multi-phase materials with the phase volume fractions varing graduall in space in a pre-determined profile. his results in continuousl graded thermomechanical properties at the macroscopic structural scale. One of the advantages of the FGM is that it can be able to withstand high-temperature-environments. Because of their speciall-tailored thermomechanical properties the are well suited for thermal protection against large temperature gradients [ ]. Due to this superior thermomechanical propert FGM plate structures have found a wide range of applications in man industries especiall in space his paper was recommended for publication in revised form b Editor Yeon June Kang * Corresponding author. el.: Fa.: address: sandzhang@ahoo.com KSME & Springer 11 vehicles and aircrafts where the are ver often subjected to high levels of thermal and dnamic loading such as large temperature gradients and acoustic pressure. his ma result in complicated large amplitude nonlinear vibration behavior of the FGM plate due to the bending stretching coupling and combined eternal loads [4]. With the increased use of these materials for structural components in man engineering applications it is necessar for us to understand the nonlinear dnamic characteristics of functionall graded plates in thermal environments. here are man studies for isotropic or laminated composite plate and shell structures such as Refs. [5-11]. Among the research about the nonlinear dnamic behaviors of the FGM plates under thermo-mechanical environment available Praveen and Redd [] adopting finite element procedure analzed the nonlinear dnamic response of functionall graded ceramic metal plates subjected to mechanical and thermal loads. Sundararajan [] studied the free vibration characteristics of functionall graded material (FGM plates subjected to thermal environment. emperature field was assumed to be a uniform distribution over the plate surface and varied in the thickness direction. Yang et al. [4] presented the

2 168 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~1646 Fig. 1. he model of a FGMs rectangular plate and the coordinate sstem. large amplitude vibration of pre-stressed functionall graded material laminated plates that were composed of a shear deformable functionall graded laer and two surface-mounted piezoelectric actuator laers. Nonlinear governing equations of motion were derived within the contet of Redd s higherorder shear deformation plate theor to account for transverse shear strain and rotar inertia. Cheng and Batra [1] studied the stead state vibration of a simpl supported functionall graded polgonal plate with temperature independent material properties. He et al. [1] presented finite element formulation based on thin plate theor for the shape and vibration control of FGM plate with integrated piezoelectric sensors and actuators under mechanical load. he constituent materials possess temperature-dependent properties. Ng et al. [14] investigated the parametric resonance of plates based on the Hamilton s principle and the assumed mode technique. Yang and Shen [15 16] analzed dnamic response of thin FGM plates subjected to impulsive loads using Galerkin procedure coupled with modal superposition method whereas b neglecting the heat conduction effect and eamined such plates and panels based on shear deformation with temperature dependent material properties. Sills et al. [17] presented different modeling aspects and also simulated the dnamic response under a step load. B adopting Laplace transformation technique and power series method Vel and Batra [18] analzed the threedimensional thermomechanical deformations of simpl supported functionall graded plates subjected to time-dependent thermal loads on its top or bottom surface. Based on perturbation technique Huang and Shen [19] dealt with the nonlinear vibration and dnamic response of FGM plates in thermal environment. Heat conduction and temperature-dependent material properties were both considered. he temperature field considered was assumed to be a uniform distribution over the plate surface and varied in the thickness direction onl. Kim and Noda [] discussed transient displacement of FGM plates due to heat flu b a Green s function approach based on the classical laminated plate theor. Jacob [1] analzed the stead-state response of a functionall graded thick clindrical shell subjected to thermal and mechanical loads. he functionall graded shell was simpl supported at the edges and it is assumed to have an arbitrar variation of material properties in the radial direction. Redd and Cheng [] used the method of asmptotic epansion to stud the three-dimensional thermoelastic deformations of functionall graded elliptic and rectangular plates. Qian and Batra [] obtained results for the steadstate and transient thermoelastic response of functionall graded plates b the meshless local Petrov Galerkin method. However to the authors knowledge the studies of the bifurcation and chaos for the FGM plates under the time-dependent thermomechanical loads have been given quite a few investigations. Since the magnitudes of transient thermal stresses are usuall larger than those of stead state stresses it is important to quantif them for proper design of an FGM plate. his paper aim focuses on a simpl supported at the fouredge FGM rectangular plate subjected to in-plane and transversal ecitation simultaneousl in the time dependent thermal environment. Here we look the FGM Plates as isotropic materials which is assumed to be temperature-dependent and graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. he geometrical nonlinearit using Von Karman s assumption is introduced. he formulation also includes in-plane and rotar inertia effects. In the framework of Redd s third-order shear deformation plate theor [4-7] the governing equations of motion for the FGM plate are derived b the Hamilton s principle. hen the equations of motion with twodegree-of-freedom under combined the time-dependent thermomechanical loads can be obtained b using Galerkin s method. Using numerical method the control equations are analzed to obtain the response curves. Under certain conditions the chaotic motions of the FGM plates are found. It is found that there eist different kinds of chaotic motions in the FGM plate.. Formulation An simpl supported at the four-edges FGM rectangular plate subjected to in-plane and transversal ecitations is considered as shown in Fig. 1. he edge width and length of the FGM rectangular plate in the and directions are respectivel a and b and the thickness is h. A Cartesian coordinate Oz is located in the middle surface of the FGM rectangular plate. Assume that ( u v w and ( u v w represent the displacements of an arbitrar point and a point in the middle surface of the FGM plate in the and z directions respectivel. It is also assumed that φ and φ respectivel represent the mid-plane rotations of two transverse normals about the and aes. he in-plane ecitation of the FGM plate is distributed along the direction at = and = a and is of the form P P 1 cosω t. he transversal ecitation subject to the FGM plate is represented b F( cosω 1t. Here Ω 1 Ω are the frequencies of the in-plane ecitation and transversal ecitation respectivel..1 FGM material properties Generall speaking most of the FGM are emploed in high-temperature environments and man of the constituent

3 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~ materials ma possess temperature-dependent properties. We assume that the temperature variation occurs with nonuniform in-plane and the stead state temperature distribution along the thickness of the plate. It is also supposed that the FGM plate is linear elastic throughout the deformation and that the plate is initiall stress free at and is subjected to a non-uniform temperature variation Δ =. he time-dependent thermal field contains two separable functions for the transient temperature variation and the spatial temperature distribution respectivel i.e. ( ( z t = + cos Ω t. (1 1 It is assumed that the plate is made from a miture of the ceramics and metals with continuousl varing such that the top surface of the plate is ceramic rich whereas the bottom surface is metal rich. he material properties P such as Young s modulus E the coefficient of thermal epansionα can be epressed as a function of the temperature see Refs. [8 9] as i ( 1 1 P = P P + + P + P + P ( 1 1 where P P 1 P 1 P and P are temperature coefficients. he effective material properties P of the FGM can be epressed as P PV t c PV b m = + ( where subscript t and b respectivel represent the top and bottom surfaces of the FGM plate V c and V m are the ceramic and metal volume fractions and add to unit Vc + V = 1. (4 m he metal volume fraction z+ h Vm ( z = h N V m is defined as where power law eponent N is a real number which characterizes the ceramic variation profile through the plate thickness. From Eqs. (-(4 the Young s modulus E the coefficient of the thermal epansion α the mass densit ρ can be epressed as = ( b t m + t ( ( V. E E E V E ρ ρ ρ ρ α α α α = b t Vm + t = b t m + t (6. heoretical equations According to the Redd s third-order shear deformation (5 theor [4-7 ] and the Hamilton s principle the nonlinear governing equations of motion for the FGM rectangular plate are given as w&& N + N = Iu + I1 c1i c1i ( && φ && (7a w&& N + N = Iv&& + ( I1 c1i && φ c1i (7b w w w w N N N N w w + N + N + c 1 P + P + P ( w u&& + N v&& = Iw && + ci 1 + && φ && φ w&& w&& + c 1( I4 ci 16 + ci 16 + (7c ( M + M c P c P Q c R 1 1 w&& = ( I 1 c1i u&& + ( I c1i4 + c1i && 6 φ c1( I4 c1i6 ( M + M c P c P Q c R 1 1 w&& = ( I 1 c1i v&& + ( I c1i4 + c1i6 && ϕ c1( I4 c1i6 where the stress resultants are given as follows: N ε N 1 N = { A B E } ε + N N ε N M ε M 1 M = { B D F } ε + M M ε M P ε P 1 P = { E F H } ε + P P ε P Q γ R γ = { A D } = { D F }. Q γ R γ (7d (7e From Eq. (8 it is known that the thermal stress resultants are represented as N Q / 11 Q1 Q h 16 α N = Q1 Q Q6 α Δ dz h / N Q61 Q6 Q 66 (8

4 164 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~1646 M Q Q Q α M = Q Q Q zδ dz M h / α h / Q61 Q6 Q 66 P Q Q Q α P = Q Q Q z Δ dz P h / α. (9 h / Q61 Q6 Q 66 Substituting the stress resultants of Eq. (9 into Eq. (7 we can write Eq. (7 in terms of generalized displacements u v w φ φ ( u u v ( A A A A ( B c E ( B c E φ φ ( B c E B c E φ w w w w w w ( A + A + A + A w w ( ce c E + E N w&& = Iu && + ( I1 ci 1 && φ ci 1 (1a v v u ( A A A A φ φ + ( B66 + c1e66 + ( B + c1e φ w w 1` ( + B c E + B c E + A w w w w ( A + A + A w w 1 1( ce c E + E N w&& = Iv && + ( I1 ci 1 && φ ci 1 (1b u w u w u w A A A u w u w ( + A + A + A u w u u ( A + c E + c E + E v w v w v w 66 ( A + A + A + A v w v w v w A + A + A v v w w 1 1( ce + c E + E + ce w ( A 55 cf55 cd55 ( c1f11 c1 H11 φ ( 4 w w A + c F c D c H w w w + c 1( E66 E1 + c 1( E1 E66 w w w w w 1 66 ( + A + A + A 4 1 w w w + A 1 + A66 c 1 H w w w w A + A A + ( 4 w φ ( w c H H + B c E φ φ + ( cd 55 + cf 55 + A 55 + ( cf 1 ch 1 ( B B ce ce φ w φ ( ( w φ w B 66 c1e66 B11 c1e11 φ w φ w ( B c E ( B c E φ + c ( F + F c H c H φ ( cd A cf ( B ce φ w φ ( w φ w + B ( 66 c1e66 + B c1e φ w φ w ( B c E ( B c E ( B B c E c E φ w φ w + c ( 1 F1 + F66 c1h1 c1h66 N w w cos w N + F Ω t N N ( w w N 1 N1 c w 1 P1 + P1 γ t

5 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~ w&& w&& u&& v&& Iw = && ci ci 1 + && φ && φ + c1( I4 c1i6 + (1c u v ( B c E ( B B c E c E u w w ( B c E ( B c E w w w + ( B11 c1e11 ( c1f11 c1 H11 ( B B ce ce w w w ( c F F + c H + c H ( A55 cd55 cf55 w + ( φ + D11 c1f 11 + c1 H11 M 1 c1p1 ( φ D ( 66 c 1 F 66 c 1 H 66 A 55 c D 55 c F 55 + ( D 1 cf 1 1 cf 1 66 D 66 ch 1 1 ch φ ( I c I u&& ( I c I c I && φ c ( I c I = v u ( B c E + ( B + B c E c E v w w ( B c E ( B c E ( B B c E c E w w ( B c1e ( c1f c1 H w w w + w ( c F F + c H + c H φ ( D c F c H + + ( A44 cd44 cf44 + w ( D 1 c 1 F 1 c 1 F 66 D 66 c 1 H 1 c 1 H 66 φ φ + ( D c1f + c1 H ( φ A c D + c F M c P φ w&& (1d ( I c I v&& ( I c I c I && φ c ( I c I = w&& (1e where γ is the damping coefficient A ij B ij D ij E ij F ij and H ij respectivel are the stiffness elements of the FGM plate which are denoted as h / 4 6 ( ij ij ij ij ij ij ij ( 1 A B D E F H Q z z z z z dz h / = ( i j = 16 (11 h / 4 ( ij ij ij ij ( 1 A D F Q z z dz = ( h / All kinds of inertias in Eq. (1 are calculated b i j = 45. (1 h i Ii = z ρ( z dz ( h N 1 Q / 11 Q1 α h N 1 = 1( cosωt Q1 Q α dz h/ N Q 66 1 L% 1 = 1( cosωt L% 1 L% 1 i = (1 M 1 Q / 11 Q1 α h M 1 = 1( Q1 Q α z dz h / M Q 66 1 L 1 = 1( cosωt L1 L 1 P 1 Q / 11 Q1 α h P1 = 1( cosωt Q1 Q α z dz h/ P Q 66 1 L 1 = 1( cos Ωt L1. (14 L1 he simpl supported boundar conditions can be epressed as at = and = a w= φ = M = N = (15a at = and = b w= φ = M = N = (15b b N ( 1cos = a d = P + P Ω t = a d. (15c b

6 164 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~1646 In order to obtain the dimensionless equations we introduce the transformations of the variables and parameters u u v = w v = w = φ = φ φ = φ = a b h a b ( ab 7 ( ab 1 1 = P1 = P1 F = F γ = b γ Eh π Eh π h ρe 1 b P = P 1 1 abρ 1 = Ω i = Eh Ωi π E ( i = 1 1 E ( ab 1 ( ab 1 t = π t A = A B = B abρ Eh Eh ( ab 1 ( ab 1 ( ab 1 D = D E = E F = F Eh Eh Eh ( ab 1 1 H = H I. 8 i = I i i (16 + Eh ab ρ ( ( 1 We mainl consider transverse nonlinear oscillations of the FGM rectangular plate in the first two modes. It is our desirable to choose a suitable mode function to satisf the first two modes of transverse nonlinear oscillations and the boundar conditions for the FGM rectangular plates. hus we write the w as follows: π π π π w w1sin sin wsin sin a b a b = + (17 where w 1 and w are the amplitudes of two modes respectivel. he transverse ecitation can be represented as π π π π F( F1sin sin Fsin sin a b a b = + (18 where F 1 and F represent the amplitude of the transverse forcing ecitation. he time dependent temperature field is defined as π π π π 1( 11sin sin 1sin sin a b a b = + (19 where 11 and 1 represent the amplitude of the temperature field. For simplicit we drop the overbars in the following analsis. Based on research given in Refs. [1 ] neglecting all inertia terms on u v φ and φ in Eq. (1 and the term of time dependent temperature stress in Eqs. (16a (16b (16d and (16e substituting Eq. (17 into Eqs. (1a (1b (1d and (1e we obtain the displacements u v φ and φ with respect to w. Substituting Eqs. (17 (18 and (19 into Eq. (1c and appling the Galerkin procedure ield the governing differential equation of transverse motion of the Fig.. Comparison of temporal evolution of center transverse deflection obtained b present results( and that published in Ref. [6] ( read from graph. FGM rectangular plate for the dimensionless as follows: w&& 1+ aw1+ a1w& 1+ aw1cosω t + aw1 + a4w + aww aw aww 7 1 ( ( + a a 9 1 w1 + a111 + a111 w + a111 cosωt = f cosω t (a 1 1 w&& + bw + bw 1& + bwcosω t + bww 1 + b4w1 + bw 5 + bww bw 7 ( ( + b b 9 1 w1 + b b11 1 w + b11 cosωt = f cos Ω t. (b 1 All coefficients aforementioned in Eq. ( are too long to be listed out in the paper for abbreviation.. Numerical simulations of periodic and chaotic motions Before proceeding to the nonlinear vibration analsis of the FGM plates a comparison eample is solved to validate the present analsis firstl. he transient response results are compared in Fig. with the finite element solutions provided b Redd [] where the temporal evolution curves of center deflection are presented for a simpl supported intact aluminum-zirconia FGM square plate ( a = b =.m h=.1m under a suddenl applied uniform load of intensit of q = 1MPa. he material composition is assumed to follow a simple power-law distribution through the thickness direction such that the plate is 1% zirconia ( Et = 151GPa ρ t = kg / m at the top surface and 1% aluminum ( Eb = 7GPa ρ b = 77 kg / m at the bottom surface. he power-law eponent and the Poisson s ratio are taken as n =. and ν =. respectivel. he dimensionless center deflection and dimensionless time are defined as w wcebh ( qa t t Eb ( ρ ba = and = respectivel. Our results agree well with the finite element results.

7 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~ (a (b (a (b (c (d (c (d (e (f (e (f Fig.. he periodic motion of the FGMs rectangular plate eists when 11 = 1 =. In the following investigation the Runge-Kutta algorithm [4] is utilized to numericall analze the periodic and chaotic motions of the FGMs rectangular plate subjected to time dependent thermal and mechanical loads. We consider the Eq. ( to do numerical simulation. o stud the thermal effect on nonlinear dnamic response we choose the time dependent temperature 11 and 1 as the controlling parameters when the periodic and chaotic responses of the FGM rectangular plate are investigated. At the same time we consider the governing equations of motion for the FGM plate without the terms of the time dependent temperature to do numerical simulation too as a comparison studies. he zirconia and titanium allo are selected for the two constituent materials of the plate in the present eamples referred to as ZrO/i-6Al-4V shown in Ref. [5]. he two-dimensional phase portrait waveform three-dimensional phase portrait are plotted to demonstrate the nonlinear dnamic behaviors of the FGMs rectangular plate. Fig. illustrates the eistence of the periodic motion for the FGMs rectangular plate when the governing equations of motion for the FGM plate don t include the terms of the time dependent temperature. Obviousl Fig. illustrates that the periodic response of the FGM rectangular plate occurs if we Fig. 4. he periodic motion of the FGMs rectangular plate eists when 11 = = don t consider the effect of the time dependent temperature. he parameters and the initial conditions are respectivel chosen as a =. 9 a 1 =.14 a = 1.8 a = 5. a 4 = 8.69 a 5 = 1.7 a 6 = 16.7 b 7 =.9 a 8 =. a 9 =.9 a 1 =.9 a 11 =.9 a 1 = 1. b = 7.8 b 1 =.14 b = b = 7.7 b 4 = 1.1 b 5 = 14. b 6 = 16.6 b 7 = 5.7 b 8 =.8 b 9 =.8 b 1 =.8 b 11 = 9. b 1 = 1. f 1 = 1.86 f = 8.79 Ω 1 = 5 Ω = 5 Ω = 1 1 =.1 =.1 =.8 4 =.16. Figs. (a and (c represent the phase portraits on the planes ( 1 and ( 4 respectivel. Figs. (b and (d respectivel denote the waveforms on the planes ( t 1 and ( t. Figs. (e and (f represent the three-dimensional phase portrait in space ( 1 and the Poincare map on plane ( 4 respectivel. Here 1 and 4 can be epressed as 1 = w 1 = w& 1 = w 4 = w& respectivel. It can be shown from Fig. that the amplitude of the second order mode is larger than one of the first order mode. With the increasing of the controlling parameters 11 and 1 multiperiodic occurs. Fig. 4 shows that the multi-periodic motion occurs when the temperature increased to 11 = 11.8 and 1 = Until the temperature is increased to 11 = 5.8

8 1644 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~1646 (a (b (a (b (c (d (c (d (e (f (e (f Fig. 5. he chaotic motion of the FGMs rectangular plate eists when 11 = = 5.8. Fig. 7. he chaotic motion of the FGMs rectangular plate eists when 11 = 1 = 85. (a (b (a (b (c (d (c (d (c (d (e (f Fig. 6. he chaotic motion of the FGMs rectangular plate eists when 11 = 18 1 = 18. Fig. 8. he chaotic motion of the FGMs rectangular plate eists when 11 = 4 1 = 4.

9 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~ (a (c (b (d oscillations of the FGM plate are considered then Galerkin s approach is utilized to discretize the governing equations of motion to a two-degree-of-freedom nonlinear sstem including the quadratic and cubic nonlinear terms. Using numerical method the control equations are analzed to obtain the response curves. Under certain conditions the chaotic motions of the FGM plates are found. It is found that because of the eistence of the temperature which relate to the time the motions of the FGM plate show the great difference. A period motion can be changed into the chaotic motions which are affected b the time dependent temperature. Acknowledgment he authors gratefull acknowledge the National Natural Science Foundation of China (NNSFC through grant No and 17 the Natural Science Foundation of Beijing (NSFB through grant No.1111 and Beijing institution of higher learning talent education program project foundation No.PHR961. References (e Fig. 9. he chaotic motion of the FGMs rectangular plate eists when 11 = 56 1 = 518. and 1 = 5.8 the response of the FGMs rectangular plate is the chaotic motion as shown in Fig. 5. From Fig. 6 which the temperature is 11 = 18 and 1 = 16 to Fig. 9 which the temperature is 11 = 8 and 1 = 88 it can illustrate that the chaotic response of the FGM rectangular plate eists. In the fact until the temperature increases to 11 = 8 and 1 = 8 the FGM plate is in the conditions of the chaotic motion. Because of the limit of the page number we don t give other figures. From Figs. -9 it can be shown that the process of change for the motions of the FGMs rectangular plate is as follows: the periodic motion the multi-periodic motion the chaotic motion. 4. Conclusions he nonlinear oscillations and chaotic dnamics of the FGMs rectangular plate under combined the transverse and inplane ecitations in the time dependent thermal environment are investigated for the first time. he materials properties are assumed to be temperature-dependent. he geometrical nonlinearit using Von Karman s assumption is introduced. Based on the Redd s third-order plate theor the governing equations of motion for the FGM rectangular plate are derived b using the Hamilton s principle. Onl transverse nonlinear (f [1] J. Sladek V. Sladek C. Hellmich and J. Eberhardsteiner Heat conduction analsis of -D aismmetric and anisotropic FGM bodies b meshless local Petrov Galerkin method Computational Mechanics original paper (6. [] G. N. Praveen and J. N. Redd Nonlinear transient thermoelastic analsis of functionall graded ceramic-metal plates International Journal of Solids and Structures 5 ( [] N. Sundararajan. Prakash and M. Ganapathi Nonlinear free fleural vibrations of functionall graded rectangular and skew plates under thermal environments Finite Elements in Analsis and Design 4 ( [4] J. Yang S. Kitipornchai and K. M. Liew Large amplitude vibration of thermo-electro-mechanicall stresses FGM laminated plates Computer Methods Applied Mechanics Engineering 19 ( [5] W. P. Chang and S. M. Wan hermomechanicall coupled non-linear vibration of plates International Journal of Nonlinear Mechanics 1 (5 ( [6] W. P. Chang and S. C. Jen Nonlinear free vibration of heated rectangular plates International Journal of Solids and Structures ( ( [7] Q. Han Z. Zhang and G. Yang Chaotic motion of a nonlinear thermo-elastic elliptic plate Applied Mathematics and Mechanics (9 ( [8] X. Shu X. Zhang and J. Zhang hermoelastic free vibration of clamped circular plate Applied Mathematics and Mechanics 1 (6 ( [9] W. Zhang and Z. M. Liu Global Dnamics of a parametricall and eternall ecited thin plate Nonlinear dnamics 4 ( [1] Y. L. Yeh and C. Y. Lo Chaotic and bifurcation dnamics of a thermo-elastic aismmetric circular plate in large de-

10 1646 Y. X. Hao et al. / Journal of Mechanical Science and echnolog 5 (7 (11 167~1646 flection Journal of Chinese Societ of Mechanical Engineering ( ( [11] Y. L. Yeh C. K. Chen and H. Y. Lai Chaotic and bifurcation dnamics for a simpl supported rectangular plate of themo-mechanical coupling in large deflection Chaos Solitons & Fractals 1 ( [1] Z. Q. Cheng and R. C. Batra hree dimensional thermoelastic deformations of a functionall graded elliptic plate Composites: Part B: Engineering 1 ( ( [1] X. Q. He. Y. Ng S. Sivashankar and K. M. Liew Active control of FGM plates with integrated piezoelectric sensors and actuators International Journal of Solids and Structures 8 ( [14] K. M. Liew X. Q. He. Y. Ng and S. Sivashankar Active control of FGM plates subjected to a temperature gradient: modeling via finite element method based on FSD International Journal for Numerical Methods in Engineering 5 ( [15] J. Yang and H. S. Shen Dnamic response of initiall stressed functionall graded rectangular thin plates Composite Structures 54 ( [16] J. Yang and H. S. Shen Vibration characteristic and transient response of shear-deformable functionall graded plates in thermal environments Journal of Sound and Vibration 55 ( [17] L. B. Sills R. Eliaso and Y. Berlin Modeling of functionall graded materials in dnamic analses Composites: Part B:ngineering ( [18] S. S. Vel and R. C. Batra hree-dimensional analsis of transient thermal stresses in functionall graded plates International Journal of Solids and Structures 4 ( [19] X. L. Huang and H. S. Shen Nonlinear vibration and dnamic response of functionall graded plates in thermal environments International Journal of Solids and Structures 41 ( [] K. S. Kim and N. Noda A Green s function approach to the deflection of a FGM plate under transient thermal loading Archive of Applied Mechanics 7 ( [1] L. Jacob Pelletier and S. S. Vel An eact solution for the stead-state thermoelastic response of functionall graded clindrical shells International Journal of Solids and Structures 4 ( [] J. N. Redd and Z. Q. Cheng hree-dimensional thermomechanical deformations of functionall graded rectangular plates European Journal of Mechanics A/Solids ( [] L. F. Qian and R. C. Batra ransient thermoelastic deformations of a thick functionall graded plate Journal of hermal Stresses 7 ( [4] J. N. Redd Mechanics of Laminated Composite Plates and Shells: heor and Analsis CRC Press Boca Raton Florida USA (4. [5] J. N. Redd A simple higher-order theor for laminated plates Journal of Applied Mechanics 51 ( [6] J. N. Redd A refined nonlinear theor of plates with transverse shear deformation International Journal of Solids and Structures ( [7] W. Aliaga and J. N. Redd Nonlinear thermoelastic response of functionall graded plates using the third-order plate theor International Journal of Computational Methods in Engineering Science and Mechanics 5 (4 ( [8] S. C. Pradhana C.. Loa K. Y. Lama and J. N. Redd Vibration characteristics of functionall graded clindrical shells under various boundar conditions Applied Acoustics 61 ( [9] Y. S. ouloukian hermophsical properties of high temperature solid materials Mac Millian New York USA (1967. [] Y. X. Hao L. H. Chen W. Zhang and J. G. Lei Analsis on Nonlinear Dnamics of Functionall Graded Materials Plate Journal of Sound and Vibration 1 (4-5 ( [1] A. Bhimaraddi Large amplitude vibrations of imperfect antismmetric angle-pl laminated plates Journal of Sound and Vibration 16 ( [] A. Nosier and J. N. Redd A stud of non-linear dnamic equations of higher-order deformation plate theories International Journal of Non-Linear Mechanics 6 ( [] J. N. Redd Analsis of functionall graded plates International Journal for Numerical Methods in Engineering 47 (1- ( [4]. S. Parker and L. O. Chua Practical numerical algorithms for chaotic sstems Springer-Verlag New York USA (1989. [5] H. S. Shen Nonlinear bending response of functionall graded plates subjected to transverse loads and in thermal environments International Journal of Mechanical Sciences 44 ( W. Zhang received his Ph.D. in 1997 from Department of Mechanics ianjin Universit and Post Doctoral Fellow in 1999 from Universit of Western Ontario Canada. He is a director of Dnamics and Control Division and a fellow for the Chinese Societ of heoretical and applied Mechanics. His research interests include the global bifurcations and chaotic dnamics of high-dimensional nonlinear sstems the global dnamics of nonlinear continuous sstems and nonlinear dnamics of mechanical sstems induced b flow.