Analysis of Functionally Graded Plates Subjected to Hygrothermomechanical Loads. Citation AIAA Journal, 2016, v. 54 n. 11, p.

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1 itle Analysis of Functionally Graded Plates Subjected to Hygrothermomechanical Loads Author(s Wu, Z; Ma, Y; Ren, XH; Lo, SH itation AIAA Journal, 6, v. 54 n., p Issued Date 6 URL Rights AIAA Journal. opyright American Institute of Aeronautics and Astronautics, Inc.; his work is licensed under a reative ommons Attribution-Nonommercial-NoDerivatives 4. International License.

2 Analysis of functionally graded plates subjected to hygro-thermo-mechanical loads Wu Zhen, Ma Yuting, Ren Xiaohui, S. H. Lo * Key Laboratory of Liaoning Province for omposite Structural Analysis of Aerocraft and Simulation, Shenyang Aerospace University, Shenyang 6, hina Department of ivil Engineering, University of Hong Kong, Pokfulam Road, Hong Kong, hina Abstract: Analytical solution taking into account the transverse normal strain is developed for the analysis of the mechanical and hygrothermal behaviors of functionally graded material (FGM plates. In order to include the transverse normal deformation, the hygrothermal deformation through the thickness due to thermal and moisture loads is introduced in the transverse displacement field. Employing the bounding surface free traction condition, effects of transverse normal hygrothermal deformation can be incorporated in the in-plane displacement field. Moreover, there are only five displacement variables in the proposed model as transverse normal deformations due to hygrothermal loads are absorbed in the generalized force vector. By analyzing the hygrothermal behaviors of FGM plates, the accuracy and efficiency of the proposed model are assessed. Moreover, the effects of temperature, moisture concentration, transverse normal strain, volume fraction distributions and side-to-thickness ratios on the displacements and the stresses of FGM plates have all been studied. Keywords: Analytical model; functionally graded plate; transverse normal strain; hygro-thermo-mechanical loads. * orresponding author. el.: address: hreclsh@ hku.hk (S.H. Lo

3 . Introduction he functionally graded materials (FGMs is generally composed of metal and ceramic in which metal provides the toughness and strength at low temperature while ceramic can offer a high resistance to heat at elevated temperature. Most FGM structures are subjected to changing environmental conditions during both initial fabrication and service life. As a result, internal stresses in FGM structures are generated with the change of hygrothermal environment, which will induce large deformation and even cause failure of the structures. Displacements and stresses in the functionally graded material structures subjected to mechanical and hygrothermal loads have been widely studied in the past, which are still the subject of research interest in recent years [-]. Based on the first-order shear deformation theory and von Karman's assumptions, Sundararajan et al. [4] studied the free vibration of FGM plates subjected to thermal load. In terms of the first-order theory, Nguyen-Xuan et al. [5] proposed an edge-based smoothed finite element method for static, free vibration and buckling analysis of FGM plates, in which thermal and mechanical properties of FGM plates are assumed to vary through the thickness by a power rule of the volume fraction of the constituents. Based on the first-order theory, Singha et al. [6] proposed a four-node quadrilateral plate bending element for the analysis of nonlinear behaviors of functionally graded plates subjected to transverse distributed loads. Nguyen-Xuan et al. [7] developed an improved finite element approach based on the first-order theory to study the mechanical/thermal buckling of FGM plates. ransverse shear strains in the first-order shear deformation theory are assumed to be constant through the thickness direction. hus, the correction factors have to be employed to adjust the transverse shear stiffness. It is noted that the accuracy of results obtained from the first-order theory is strongly influenced by the shear correction factors [8,9]. In order to represent the smooth distribution of transverse shear stresses across the thickness, the higher-order shear deformation models have been developed for the thermomechanical analysis of the FGM plates.

4 In order to study the effects of volume fractions and modulus ratio of the constituents on displacements and transverse shear stresses of FGM plates, Reddy [] presented a theoretical formulation and a finite element model based on a third-order shear deformation []. Analytical solution in terms of the third-order model [] allows parabolic variation of transverse shear strain along the plate thickness and satisfies the transverse shear free conditions on the plate surfaces. Zenkour [] employed a sinusoidal shear deformation theory (SP [] to study the static behaviors of a simply-supported functionally graded plates subjected to a transverse uniform loads. he sinusoidal shear deformation theory satisfies the traction-free boundary conditions and transverse shear correction factors are not required. On the other hand, Zenkour and Alghamdi [4] extended the model SP to study the thermoelastic bending behaviors of functionally graded sandwich plates. Employing the sinusoidal shear deformation plate theory, Zenkour [5] investigated the thermoelastic bending behaviors of a simply-supported FGM plate subjected to thermomechanical loading and resting on elastic foundation. Moreover, effects of side-to-thickness ratio, volume fraction distributions and elastic foundation parameters are studied in detail. Subsequently, the model SP was extended to investigate hygrothermal bending of functionally graded material plates resting on elastic foundations [6]. Applying the sinusoidal shear deformation plate theory in conjunction with the first-order, the third-order, and the classical plate bending theories, Zenkour and Alghamdi [7,8] studied the flexural behaviors of functionally graded sandwich plates subjected to mechanical and thermal loads. Based on the sinusoidal shear deformation plate models, Zenkour et al. [9] studied the bending behaviors of a simply-supported functionally graded viscoelastic sandwich beam resting on Pasternak's elastic foundation. Zenkour and Sobhy [] further employed the sinusoidal shear deformation plate model and the third-order model to evaluate the static response of simply-supported FGM viscoelastic sandwich plates under transverse uniform loads. Employing the sinusoidal shear deformation plate theory, Zenkour and Sobhy [] computed the deflection and stresses in functionally graded plates resting on two-parameter elastic foundations. based on higher-order theories

5 with five displacement variables, Sobhy [] investigated the vibration and buckling behaviors of exponentially graded material sandwich plates resting on elastic foundations. By means of the generalized shear deformation models including the third-order model [], the sinusoidal shear deformation plate model [], the fifth-order shear deformation model ( [], trigonometric shear deformation model [4], exponential shear deformation model (ESD [5], hai et al. [6] studied the static, dynamic and buckling behaviors of functionally graded material sandwich plates. It is seen from the literature that the generalized shear deformation models with five displacement parameters are widely used to study the hygrothermal behaviors of the FGM structures [-6]. However, the transverse normal strain has been neglected in these models. For thermomechanical problems even in the moderate thick plate configuration, transverse normal deformation effect ought to be considered as the effect of out-of-plane thermal deformation is equally important in comparison with those of the in-plane thermal deformation [7-9]. Due to the omission of transverse normal strain, the third-order model with five displacement variables failed to produce accurate results for thermal expansion problems []. o extend the generalized higher-order model for hygrothermal analysis of FGM plate, a refined model including transverse normal strain will be developed. Although transverse normal strain is taken into account, there are only five displacement parameters in the proposed model as transverse normal deformations due to hygrothermal loads can be absorbed in the generalized force vector. In the proposed model, the equations of equilibrium are established by means of the principle of minimum total potential energy. o maintain the simplicity of the analytical solution with the same number of displacement parameters, the main idea is to extend the Reddy's solution to simply-supported FGM plates subjected to hygro-thermo-mechanical loads by applying the Navier's technique []. Several numerical examples are presented to demonstrate the reliability and accuracy of the proposed model in comparison with other existing higher-order models in the literature. 4

6 . heoretical formulation A rectangular FGM plate with length a, width b and thickness h is considered. he FGM plate is subjected to transverse loading q(x, y, ±h/ on the surfaces, a temperature field (x, y, z and a moisture concentration (x, y, z. he material properties Μ of the FGM plate, such as Young's modulus E, thermal expansion coefficient α, and moisture expansion coefficient β are given by [] z Μ Μ metal + Μceramic Μ metal + h ( z (.5 k ( where Μ metal and Μ ceramic represent respectively the corresponding material properties of the metal and the ceramic, and k denotes the volume fraction exponent. he mixture of metal and ceramic with continuously varying volume fraction can avoid the interface problems of laminated composites.. Reddy-type higher-order plate theory considering transverse normal hygrothermal strain for the FGM plate ( In order to model accurately the hygrothermal behaviors of the FGM plate, a Reddy-type higher order shear deformation plate model taking into account the transverse normal strain ( is proposed in the present work. o include transverse normal effect which is significant for the hygrothermal problems, transverse deformation due to hygrothermal loading is introduced in the out-of-plane displacement field. hus, the initial displacement field is given by u u + zu + z u + z u v v + zv + z v + z v w w + w + w ( where w is the transverse normal deformation due to thermal load and w is the transverse normal deformation caused by moisture concentration load, which are given by (,, α (,, w x y z x y z dz z (,, β (,, w x y z x y z dz z ( 5

7 where α z and β z are respectively the transverse normal thermal expansion coefficient and the transverse normal moisture expansion coefficient through the thickness of the plate. he temperature field (x, y, z and the moisture concentration (x, y, z are given by (,, ( (, (,, ( (, x y z f z x y x y z f z x y (4 Employing equations ( and (4, transverse normal deformation due to hygrothermal load can be written as (,, F ( (, (,, F ( (, w x y z z x y w x y z z x y (5 where F F ( α ( z f z dz z ( β ( z f z dz z he transverse shear stresses related linearly with the shear strains are given by ( z ( z τ xz D44 γ xz (7 τ yz D55 γ xz E( z D44 z D55 z, E(z and υ are Young's modulus and Poisson's + υ where, ( ( ( ratio of the FGM plate, respectively. From equation (, we have (6 γ γ xz yz w w w u+ zu + z u+ + + x x x w w w v+ zv + z v+ + + y y y (8 Employing the top and the bottom surface transverse shear free condition, we have w w w γ xz z± h/ u + zu + z u x x x w w w γ yz z± h/ v + zv+ z v y y y (9 Applying the zero shear stress boundary conditions at the top and the bottom surfaces, a Reddy-type higher-order plate theory including transverse normal 6

8 hygrothermal strain ( is given by w u u +Φ u+φ +Φ +Φ4 x x x w v v +Ψ v+ψ +Ψ +Ψ4 y y y x w w + w + w ( where 4z h 4z Φ, h Φ z, ( z ( z 4z ( z + ( z z F F F F Φ, h h ( z ( z 4z ( z + ( z z F F F F Φ 4 ; h h 4z h 4z Ψ, h Ψ z, ( z ( z 4z ( z + ( z z F F F F Ψ, h h ( z ( z 4z ( z + ( z z F F F F Ψ 4. h h where z and z are respectively the z-coordinate at the bottom and the top of the ply as shown in Figure. In equation (, it is found that there are only five displacement variables in the displacement field of the proposed model. In order to assess the performance of the model, the generalized higher-order shear deformation theories with five displacement parameters [] is chosen for comparison, which is given by u x y z w u x y z z u x x y v x y z w v x y z z v y x y (,, (, + Ψ ( (, (,, (, + Ψ ( (, (,, w( x, y w x y z ( 7

9 A Reddy-type higher-order shear deformation plate theory neglecting transverse normal strain ( [] can be given by setting function Ψ(z as Ψ ( z 4 z z h ( In addition, a displacement field of the sinusoidal shear deformation plate theory (SP [] can be given by setting Ψ h π z π h ( z sin ( Karama et al. [5] proposed a higher-order shear deformation theory ESD in which the function Ψ(z is set as Ψ ( z ( z/ h ze (4 A higher-order shear deformation theory [] has been proposed by setting function Ψ(z as 5 Ψ 7 z z ( z z 4 8 h + h (5 A ninth-order theory proposed by Matsunaga [] will be employed to assess the performance of the generalized higher-order shear deformation models with five displacement parameters. he in-plane displacement field of the model consists of 9th-order polynomial in the global thickness coordinate z and the transverse deflection is represented by an 8th-order polynomial in z. Performance of the model has been verified in reference [], so results of will be used as reference solutions to assess the performance of all other models. he ninth-order theory is given by 8

10 9 i u( x, y, z u ( x, y + z ui i 9 i v( x, y, z v ( x, y + z vi i 8 i w( x, y, z w( x, y + z wi i (6 In equation (6, it is found that there are 9 variables in the displacement field of the mode.. onstitutive equations For linear elasticity, the strain components of the model can be written as u u w ε x +Φ +Φ +Φ +Φ 4 x x x x x v v w ε y +Ψ +Ψ +Ψ +Ψ 4 y y y y y γ γ γ xy u u v v w +Φ + +Ψ + ( Φ +Ψ y y x x x y + Φ +Ψ + Φ +Ψ xy ( ( 4 4 xy Φ Φ w Φ z z x z x xz u Φ4 w w + + z x x x Ψ Ψ w Ψ Ψ w w z z y z y z y y y yz v (7 he temperature distribution and the moisture concentration through the thickness of plate are taken from reference [6] (,, (, + ( / (, + ( z / (, (,, (, + ( / (, + ( / (, x y z x y z h x y f h x y x y z x y z h x y f h x y where f ( h/ π sin ( πz/ h z z. (8 he relationships between stresses and strains of the FGM plate subjected to hygrothermal load can be written as 9

11 ( ( ( ( σ x D z D z ε x σ y D z D z ε y τ xy D ( z γ xy τ D ( z γ xz 44 xz τ yz D55 ( z γ yz (9 where ε ε α( Δ β( Δ, ε ε α( β( γ x x z z xz γ xz, γ yz γ yz D D44 D55 y y z Δ z Δ, γ xy γ xy, E( z υe( z ; D ( z, D ( z D ( z, D ( z E( z ( + υ υ υ ( E z, υ ; α is the linear thermal expansion coefficients in the plate coordinates; β is the moisture expansion coefficients; Δ is the rise of temperature with respect to the reference temperature, where is the reference temperature; Δ is the rise of moisture concentration with respect to the reference moisture concentration.. Equilibrium equations Applying the principle of minimum potential energy, the equilibrium equations for the model can be established. Rectangular artesian coordinates (x, y, z are used to describe the deformation of the FGM plate, in which x [x, x a ] represents the plate longitudinal axis, y [y, y b ] represents the plate width axis, and z [-h/, h/] is the thickness coordinate. he principle of minimum potential energy can be expressed as δ U δw ( where the virtual strain energy δ U and virtual work δ W done by applied loads are respectively given by where ( x x y y xy xy xz xz yz yz h/ dzdxdy A h/ ( δu σ δε + σ δε + τ δγ + τ δγ + τ δγ δw δw + δw + δw + δw + δw + δw + δw px py q x xa y yb

12 A δw p δu( x, y, z + p δu( x, y, z dxdy b t px A x x δw p δv( x, y, z + p δv( x, y, z dxdy b t py A y y b t δwq q δw( x, y, z + qδw( x, y, z dxdy h/ [ (,, (,, ] δw δu x y z + δw x y z dz x h/ x xz h/ δw δv( x, y, z + δw( x, y, z dz y h/ y yz ( [ (,, (,, ] h/ xa / xa a + h xza a h/ yb h/ yb b + yzb b δw δu x y z δw x y z dz δw δv( x, y, z δw( x, y, z dz where p b (x,y and p t (x,y are the distributed loads on the bottom and top surfaces of the plate; q b (x,y, q t (x,y are the transverse loads on the bottom and the top surfaces of the plate; x, xa, y and yb are the axial loads at the end cross-sections of the FGM plate; xz, xza, yz and yzb are the transverse shear tractions at the end cross-sections of the plate. By employing integration by parts and collecting the variational coefficients δ u, δ u, v given by δ, δ v and δ w, the equilibrium equations for the model are δ u : N + N + p x, x xy, y x δ u : M + M V + p xφ, x xyφ, y φ x δ v : N + N + p y, y xy, x y where δ v : M + M V + p ( yψ, y xyψ, x ψ y δ w : M + M + M + M xφ, xx yψ, yy xyφ, xy xyψ, xy V V + p + p q φ, x ψ, y x, x y, y h/ k k k,, x xφ xφ σ, / x σx, h σ x N M M Φ Φ dz; h/ k k k,, y yφ yφ σ, / y σ y, h σ y N M M Ψ Ψ dz;

13 V V Φ Φ z z z ; h/ k k [, ], d φ φ τ τ h/ xz + xz V V Ψ Ψ z z z ; h/ k k [, ], d ψ ψ τ τ h/ yz + yz h/ k k k k k,,,, xy xyφ xyφ xyψ xyψ τ, / xy τxy, τxy, τxy, h τ xy N M M M M Φ Φ Ψ Ψ dz; [ p, p, p ] [ p +p, Φ p + Φ p, Φ p + Φ p ] b t b N t b N t x x x x x x x x x [ p, p, p ] [ p +p, Ψ p + Ψ p, Ψ p + Ψ p ] b t b N t b N t y y y x x x x x x [ ] [ b t q q +q ]. A set of consistent geometric (kinematic-variable and kinetic (stress-resultant boundary conditions [] at the boundaries of the plates (xx, xx a, yy and yy b can be expressed as u ( x, y u or N ( x, y N β β x β xβ u( xβ, y uβ or Mx ( x, y Mx φ β φ β v ( x, y v or N ( x, y N λ λ y λ yλ v ( x, y v or M ( x, y M β λ yψ λ yψ λ w ( x, y w or M ( x, y M, x β, xβ xφ β xφ β w ( x, y w or M ( x, y M (4, y λ, yλ w( xβ, y wxβ or yψ λ yψ λ V ( x, y M ( x, y M ( x, y p ( x, y V φ β xφ, x β xyφ, y β x β xxβ w( x, yλ wyλ or V ( xy, M ( xy, M ( xy, + p ( xy, V ψ λ yψ, y λ xyψ, x λ y λ yxλ where x x λ y y ; β, a ;, b h/ [ Nxβ, Mφβ, Mφβ] [, / x x, x ]dz h β Φ β Φ β ; h/ [ N, M, M ] [, Ψ, Ψ ]d z yλ ψλ ψ λ h/ yλ yλ yλ

14 h/ [ V ] [ ]dz, ( β, a ; xxβ h/ h/ h/ xzβ [ V ] [ ]dz, ( λ,b ; yxλ yzλ.4 Analytical solution Analytical solutions in terms of trigonometric functions for the FGM plates with simply-supported boundaries are considered. he external force, the temperature loads and moisture concentration loads are given by q q sin( α xsin( β y sin( αxsin( βy i i sin( α xsin( β y i i (5 where i,,. he simply-supported boundary conditions along the edges of the FGM plate are given as follows: At edges x and xa ; v, v, w (6 At edges y and yb; u, u, w (7 Following the Navier s procedure [], analytical solution in terms of the displacement parameters satisfying the simply-supported boundary conditions are expressed as a double trigonometric series, such that u u cosα xsin β y mn m n u u cosα xsin β y mn m n v v sinα xcos β y mn m n v v sinα xcos β y mn m n (8 w w sinα xsin β y mn m n

15 where α m π / a ; β n π / b ; a and b denote respectively the length and the width of plates. Substituting equations (8 into equilibrium equations ( and collecting coefficients, the following system equation can be obtained. K U P ( where displacement parameters { u u v v w } U. In equation (9, it is found that there are only five displacement variables in the model as transverse normal deformations w and w due to hygrothermal loads have been absorbed in the generalized force vector P 5.. Results and discussion In this section, the static behavior of aluminum and alumina FGM plates subjected to hygro-thermo-mechanical loads is investigated by using the proposed model and other generalized higher-order models. he top surface of the FGM plate is ceramic (alumina rich and the bottom surface is metal (aluminum rich, which can be seen in Figure. Young's modulus, Poisson's ratio and coefficient of thermal expansion for aluminum are respectively 7 GPa,., ( -6 /, and for alumina are 8 GPa,. and 7.4( -6 /, respectively [8]. he effective Young's modulus E(z, thermal expansion coefficient α(z and moisture expansion coefficient β(z in the thickness direction of FGM plate can be computed using equation (. Example :Bending of a FGM square (ab plate subjected to thermal loading z ( x, y sin ( πx/ a sin ( πy/ b. h ransverse displacement is expressed in terms of the following dimensionless parameter. ( /, /, wa b z wˆ, α c / hα c ransverse displacements obtained using various models are presented in able. It is remarked that results obtained from the models, SP, ESD, and 4

16 are recalculated in this paper. Numerical results showed that the transverse displacements obtained using the model for the FGM plate subjected to thermal loading are in excellent agreement with those (* obtained in reference [] based on a similar analysis. As a result, the model will be employed to assess the proposed model as well as other generalized higher-order models. Example : he FGM plates subjected to hygro-thermo-mechanical loads qq sin(πx/asin(πy/b, ( x, y, z ( x, y ( z/ h ( x, y ( f / h ( x, y (,, (, ( / (, ( / (, z + + and z x y z x y + z h x y + f h x y, where ( x, y, ( x, y, ( x, y (,, sin ( πx/ a sin ( πy/ b, ( x, y, ( x, y, ( x, y (,, sin ( x / a sin ( y/ b he dimensionless displacements and stresses are given by π π. (, /, u b z u hα a, w wa ( /, b/, z, σ σ ( a/, b/, z hα x Eα c x, τxy τxy,, Eα c Eα ( z, τ τ (, b/, z xz c xz, E c 8 GPa, α -6 / his example illustrates the effects of temperature and moisture concentration on the displacements and the stresses of FGM plate. Figure depicts the distributions of displacements and stresses through the thickness of the FGM plates subjected to hygro-thermo-mechanical loads (q,,, -,, k. It is found that the results obtained from the model are in good agreement with those computed from the model for thick (a/h and moderately thick (a/h4 FGM plates. However, the generalized higher-order theories, SP, ESP and are less accurate as transverse normal strain has been neglected in these models. For thin (a/h FGM plate, all models produce almost the same displacements and stresses. Numerical results show that hygrothermal response of thin FGM plates is insensitive to transverse normal strain, while 5

17 transverse normal strain has significant impact on hygrothermal behaviors of thick FGM plates. In Figure, the effect of volume fraction exponent k on the displacements and stresses of the FGM plate subjected to hygro-thermo-mechanical load (q,,, -,, a/h4 has been studied. he FGM plates are very sensitive to the change of volume fraction exponent k. he maximum absolute values of all results increase with the increase of the volume fraction exponent k. Numerical results also show that the distribution of in-plane displacements obtained from the models, SP, ESP and is linear, while in-plane displacements obtained from the model are not linear across the thickness of the FGM plate. In addition, the model produces the minimum in-plane displacements near the middle surface of the FGM plate, while the models, SP, ESP and yield the minimum in-plane displacements at the bottom surface of the plate. It ought to be indicated that transverse shear stresses are obtained by integrating three-dimensional equilibrium equation through the thickness of plates. However, transverse shear stresses obtained from the models, SP, ESP and deviate substantially from those of as shown in Figure. In addition, results obtained from the models, SP, ESP and are closer to those computed from the models and with an increase of volume fraction exponent k. Figure 4 depicts the effects of loading on the displacements and the stresses of the thick FGM plates (a/h. For hygrothermal expansion of FGM plate (q,,, -,, the models and are pretty bad in producing reliable displacements and stresses. In terms of the models and, the minimum in-plane stresses In fact, the minimum in-plane stresses σ x occur at the bottom surface of plate. σ x obtained from the models and HSD98 appear close to the middle surface. In addition the models and produce the minimum in-plane displacements near the middle surface of the FGM plate, while the models and yield the minimum in-plane 6

18 displacements at the top surface of the plate. Furthermore, transverse shear stresses obtained from the models and are close to zero, which completely differs from those computed from the models and. With increase of hygrothermal loads, in-plane displacement u and in-plane shear stress τ xy at the top surface of plate rapidly increase, whereas in-plane displacement and in-plane shear stress at the bottom surface of the plate rapidly decrease. However, transverse shear stresses are insensitive to hygrothermal loads. Figure 5 shows the effects of the aspect ratio a/b on the displacement and the stresses of the FGM plate. he in-plane displacement u and the in-plane shear stress τ xy gradually decrease as the aspect ratio a/b increases. However, the absolute values in-plane stresses shear stresses τ xz increase with the increase of the aspect ratio a/b. σ x and transverse 4. onclusion A analytical model in terms of five displacement parameters has been developed for hygro-thermo-mechanical analysis of FGM plates. ompared to other generalized higher-order theories with five displacement parameters, the model takes into account the transverse normal strain without adding any extra degrees of freedom. Based on the proposed model, the equations of equilibrium have been developed by using the principle of minimum total potential energy. Analytical solutions for the simply-supported FGM plates subjected to hygro-thermo-mechanical loads are presented by applying the Navier procedure. Relative to the other models of the same number of displacement parameters, numerical results show that the model agrees particularly well with the analytical solution of the model with 9 displacement variables. he influence of the temperature, moisture concentration, volume fraction, the thickness-to-side ratio and the aspect ratio on displacements and the stresses is very significant as shown in the case of the simply-supported FGM plates. More importantly, it is found that distributions of displacements and stresses through the thickness obtained from the proposed model generally differ 7

19 from those obtained from other generalized higher-order models of thick FGM plates. From the computed results, it is found that the proposed model is very promising to offer a simple and effective solution to FGM plate structures subjected to hygro-thermo-mechanical loads. Acknowledgement he work described in this paper has been supported by the Seed Funding Programme for Basic Research for the project "Modelling and analysis of curved composite laminated shell structures" (4598 of the University of Hong Kong and the National Natural Sciences Foundation of hina [No. 77, 45]. References [] Zenkour AM, Exact solution of thermal stress problem of an inhomogeneous hygrothermal piezoelectric hollow cylinder. Applied Mathematical Modelling, 4; 8:6-64. [] Kar VR, Panda SK, hermoelastic analysis of functionally graded doubly curved shell panels using nonlinear finite element method. ompos Struct, 5; 9:-. [] aibi FZ, Benyoucef S, ounsi A, Bouiadjra RB, Bedia EAA, Mahmoud SR, A simple shear deformation theory for thermo-mechanical behaviour of functionally graded sandwich plates on elastic foundations. Journal of Sandwich Structures and Materials, 5; 7:99-9. [4] Sundararajan N, Prakash, Ganapathi M, Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environment. Finite Elem Anal Des, 5; 4:5-68. [5] Nguyen-Xuan H, ran LV, Nguyen-hoi, Vu-Do H, Analysis of functionally graded plates using an edge-based smoothed finite element method. ompos Struct, ; 9:9-9. 8

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23 y h / h / x q t ( x,y t ( x,y t ( x,y yz z z xa p t x xza z ( x, y p b x xz x p t y ( x, y b p y ( x, y ( x, y eramic Metal y yzb yb Fig. A FGM plate subjected to various loads

24 SP ESP a / h (, /, u b z SP ESP a / h 4 (, /, u b z SP ESP a / h (, /, u b z SP ESP.4... SP ESP.4... SP ESP -. a / h -. a / h 4 -. a / h ( /, /, σ x a b z ( /, /, σ x a b z ( /, /, σ x a b z

25 SP ESP a / h SP ESP a / h SP ESP a / h (,, τ xy z (,, τ xy z (,, τ xy z SP ESP SP ESP SP ESP -. a / h -. a / h 4 -. a / h (, /, τ xz b z (, /, τ xz b z (, /, τ xz b z Fig. omparison of displacement and stresses through thickness of the FGM plate subjected to thermomechanical loading (k, q,,, -, 4

26 k.5 SP ESP k.8 SP ESP k SP ESP (, /, u b z (, /, u b z (, /, u b z SP ESP.4... SP ESP.4... SP ESP -. k.5 -. k.8 -. k ( /, /, σ x a b z ( /, /, σ x a b z ( /, /, σ x a b z

27 SP ESP.4... SP ESP.4... SP ESP -. k.5 -. k.8 -. k (,, τ xy z (,, τ xy z (,, τ xy z k k k SP ESP (, /, τ xz b z (, /, τ xz b z (, /, τ xz b z Fig. omparison of displacement and stresses through thickness of the FGM plate subjected to thermomechanical loading (a/h4, q,,, -, 6

28 (, /, u b z q (, /, u b z q q (, /, u b z q ( /, /, σ x a b z ( /, /, σ x a b z q ( /, /, σ x a b z q

29 (,, τ xy z q q (,, τ xy z (,, τ xy z q q (, /, τ xz b z q (, /, τ xz b z q (, /, τ xz b z Fig. 4 omparison of displacement and stresses through thickness of the FGM plate subjected to various loadings (a/h, k.8 8

30 a / b a / b a / b (, /, u b z (, /, u b z (, /, u b z a / b a / b a / b ( /, /, σ x a b z ( /, /, σ x a b z ( /, /, σ x a b z

31 a / b.8. a / b.5. a / b (,, τ xy z (,, τ xy z (,, τ xy z a / b a / b a / b (, /, τ xz b z (, /, τ xz b z (, /, τ xz b z Fig. 5 omparison of displacement and stresses through thickness of the FGM plate subjected to various loadings (a/h4, k, q,,, -

32 able ransverse displacement ŵ of the FGM plate subjected to thermal loading (q,,, a/h Models k k.5 k ESD * [] ESD * []