Static Analysis of Functionally Graded Sandwich Plates Using an Efficient and Simple Refined Theory

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1 Chinee Journal of Aeronautic 4 (11) Content lit available at ScienceDirect Chinee Journal of Aeronautic journal homepage: Static Analyi of Functionally Graded Sandwich Plate Uing an Efficient and Simple Refined Theory a,b a,c a d Hadj Henni ABDELAZIZ, Haen Ait ATMANE, Imail MECHAB, Lahdar BOUMIA, Abdelouahed TOUNSI a, *, Adda Bedia El ABBAS a a Laboratoire de Matériaux et Hydrologie, Univerité de Sidi Bel Abbe, BP 89 Cité Ben M hidi, Sidi Bel Abbe, Algérie b Département de Génie Civil, Univerité Ibn Khaldoun, BP 78 Zaaroura, Tiaret 14, Algérie c Département de Génie Civil, Faculté de Génie Civil et darchitecture, Univeité de Chlef, Chelf, Algérie d Département de Phyique, Univerité de Sidi Bel Abbe, BP 89 Cité Ben M hidi, Sidi Bel Abbe, Algérie Received 13 December 1; revied 6 February 11; accepted 1 April 11 Abtract In thi paper, a new diplacement baed high-order hear deformation theory i introduced for the tatic repone of functionally graded andwich plate. Unlie any other theory, the number of unnown function involved i only four, a againt five in cae of other hear deformation theorie. The theory preented i variationally conitent, ha trong imilarity with claical plate theory in many apect, doe not require hear correction factor, and give rie to tranvere hear tre variation uch that the tranvere hear tree vary parabolically acro the thicne atifying hear tre free urface condition. Two common type of functionally graded andwich plate, namely, the andwich with functionally graded faceheet and homogeneou core and the andwich with homogeneou faceheet and functionally graded core, are conidered. Governing equation are derived from the principle of virtual diplacement. The cloed-form olution of a imply upported rectangular plate ubjected to inuoidal loading ha been obtained by uing the Navier method. The validity of the preent theory i invetigated by comparing ome of the preent reult with thoe of the claical, the firt-order and the other higher-order theorie. It can be concluded that the propoed theory i accurate and imple in olving the tatic bending behavior of functionally graded andwich plate. Keyword: functionally graded material; andwich plate; refined plate theory; bending; Navier olution 1. Introduction1 Functionally graded material (FGM) are new inhomogeneou material which have been widely ued in many engineering applicant uch a nuclear reactor and high-peed pacecraft indutrie. The me- [1] chanical propertie of FGM vary moothly and continuouly from one urface to the other. Typically thee *Correponding author. Tel.: addre: tou_abdel@yahoo.com Elevier Ltd. Open acce under CC BY-NC-ND licene. doi: 1.116/S1-9361(11)651-4 material are made from a mixture of ceramic and metal or from a combination of different material. The ceramic contituent of the material provide the hightemperature reitance due to it low thermal conductivity. The ductile metal contituent on the other hand, prevent fracture caued by tree due to the high temperature gradient in a very hort period of time. Furthermore a mixture of ceramic and metal with a continuouly varying volume fraction can be eaily manufactured [-3]. With the development in manufacturing method [,4-7] FGM eem to have great potential in andwich tructure. The analyi of thee material ha been conidered by many reearcher.

2 No.4 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) A coniderable amount of literature exit on andwich plate a they are ued in large number of application varying from high-performance compoite in aeropace tructure to low-cot material for building contruction. The limitation of claical plate theory in decribing complex problem (e.g., contact/impact problem, behavior of thic laminate plate) neceitated the development of higher-order theorie. The term higher-order refer to the level of truncation of term in a power erie expanion of diplacement about the thicne coordinate. Reiner [8] and Midlin [9] were the firt to propoe a plate theory that included the effect of hear deformation and that aumed linear longitudinal diplacement and contant tranvere diplacement. Midlin [9] introduced the correction factor into the hear tre to account for the fact that the model predict a uniform hear tre through the thicne of the plate. Yang, et al. [1] extended Midlin theory for homogeneou plate to laminate coniting of arbitrary number of bonded layer. Baed on the ame model (Midlin theory), Whitney and Pagano [11] developed a theory for aniotropic laminated plate coniting of an arbitrary number of bonded aniotropic layer that include hear deformation and rotary inertia. Diplacement field i aumed to be linear in thicne coordinate. Since then, the plate theory wa improved by including higher-order term in diplacement aumption. Eenburg [1] aumed econd-order tranvere diplacement and linear longitudinal diplacement; Reiner [13] included third-order term in the in-plane diplacement z-expanion; Lo, et al. [14-15] included third-order in-plane and econd-order out-of-plane term. Reddy [16] developed a third-order hear deformation plate theory (TSDPT) for compoite laminate, baed on aumed diplacement field (third-order in-plane and contant out-of-plane diplacement). The FGM andwich plate can alleviate the large interfacial hear tre concentration becaue of the gradual variation of material propertie at the faceheet-core interface. The effect of FGM core were tudied by Venataraman and Sanar [17], and Anderon [18] on the hear tree at the faceheet-core of FGM andwich beam. Pan and Han [19] analyzed the tatic repone of the multilayered rectangular plate made of functionally graded, aniotropic and linear magneto-electro-elatic material. Da, et al. [] tudied a andwich compoed of a ingle FGM oft core with relatively orthotropic tiff faceheet by uing a triangular plate element. Shen [1] conidered two type of FGM hybrid laminated plate, one i with FGM core and piezoelectric ceramic faceheet and the other i with FGM faceheet and piezoelectric ceramic core. The FGM andwich contruction commonly exit in two type: FGM faceheet-homogeneou core and homogeneou faceheet-fgm core. For the cae of homogeneou core, the oftcore i commonly employed becaue of the light weight and high bending tiffne in the tructural deign. The homogeneou hardcore i alo employed in other field uch a control or in the thermal environment. The actuator and enor which are commonly piezoelectric ceramic are alway in the mid layer of the andwich contruction a in the paper of Shen [1]. Moreover, in the thermal environment, the metal-rich faceheet can reduce the large tenile tre on the urface at the early tage of cooling []. A far a we now, there ha been no invetigation on bending repone of FGM andwich plate uing two variable refined plate theory (RPT). Thi theory wa developed by Shimpi [3] for iotropic plate, and wa extended by Shimpi and Patel [4-5] for orthotropic plate. The mot intereting feature of thi theory i that it doe not require hear correction factor, and ha trong imilaritie with the claical plate theory (CPT) in ome apect uch a governing equation, boundary condition and moment expreion. The purpoe of thi paper i to extend the RPT developed by Shimpi and Patel [4-5] to the tatic repone of FGM andwich plate. The preent theory atifie equilibrium condition at the top and bottom face of the andwich plate without uing hear correction factor. Navier olution i ued to obtain the cloed-form olution for imply upported FGM andwich plate. Numerical example are preented to illutrate the accuracy and efficiency of the preent theory in predicting the tree and diplacement of FGM andwich plate by comparing the prediction with thoe computed uing variou theorie.. RPT for FGM Sandwich Plate.1. Geometrical configuration Conider the cae of a uniform thicne, rectangular FGM andwich plate compoed of three microcopically heterogeneou layer referring to rectangular coordinate (x, y, z) a hown in Fig. 1. The top and bottom face of the plate are at z=h/, and the edge of the plate are parallel to axe x and y. The two ide lengthe are a and b. Fig. 1 Geometry of rectangular FGM andwich plate with uniform thicne in rectangular Carteian coordinate. The andwich plate i compoed of three elatic layer, namely, Layer 1, Layer, and Layer 3 from bottom to top of the plate. The vertical ordinate of the bottom, the two interface, and the top are denoted by h 1 = h/, h, h 3, h 4 =h/, repectively. For the brevity, the ratio of the thicne of each layer from bottom to top i denoted by the combination of three number,

3 436 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) No.4 i.e. 1--1, -1- and o on. A hown in Fig., two type, Type A and Type B are conidered in the preent tudy: Type A FGM faceheet and homogeneou core. Type B Homogeneou faceheet and FGM core. Fig. Material variation along the thicne of FGM andwich plate... Material propertie The propertie of FGM vary continuouly due to the gradual change in the volume fraction of the contituent material, uually in the thicne direction only. Power-law function i commonly ued to decribe thee variation of material propertie. The andwich tructure made of two type of power-law FGM mentioned before are dicued a follow...1. Type A power-law FGM faceheet and homogeneou core The volume fraction of the FGM i aumed to obey a power-law function along the thicne direction: (1) z h1 V z[ h1, h] h h1 () V 1 z[ h, h3] (1) (3) z h4 V z[ h3, h4] h3 h 4 where V ( n) (n=1,, 3) denote the volume fraction function of Layer n; the volume fraction index ( +), which dictate the material variation profile through the thicne.... Type B homogeneou faceheet and powerlaw FGM core The volume fraction of the FGM i aumed to obey a power-law function along the thicne direction: (1) V z[ h1, h] () z h V z[ h, h3] () h3 h (3) V 1 z[ h3, h4] The effective material propertie, lie elatic modulu E and Poion ratio, then can be expreed by the rule of mixture [6] a ( n) ( n) P ( z) P ( P1 P) V (3) where P (n) i the effective material property of FGM of Layer n. For Type A, P 1 and P are the propertie of the top and bottom face of Layer 1, repectively, and vice vera for Layer 3 depending on the volume fraction V (n) (n=1,, 3). For Type B, P 1 and P are the propertie of Layer 3 and Layer 1, repectively. Thee two type of FGM andwich plate will be dicued later in the following ection. For implicity, Poion ratio of plate i aumed to be contant in thi tudy for that the effect of Poion ratio on the deformation i much le than that of elatic modulu [7]..3. Baic aumption Aumption of the preent refined plate theory are a follow: (1) The diplacement are mall in comparion with the plate thicne and, therefore, train involved are infiniteimal. () The tranvere diplacement W include two component of bending w b and hear w. Thee component are function of coordinate x, y only. W( x, y, z) w ( x, y) w ( x, y) (4) b (3) The tranvere normal tre z i negligible in comparion with in-plane tree x and y. (4) The diplacement U in x-direction and V in y-direction conit of extenion, bending and hear component. U uub u (5) V v vb v The bending component u b and v b are aumed to be imilar to the diplacement given by the claical plate theory. Therefore, the expreion for u b and v b can be given a wb ub z x wb vb z y (6a) The hear component u and v, in conjunction with w, give rie to the parabolic variation of hear train xz, yz, and hence to hear tree xz, yz through the

4 No.4 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) thicne of the plate, in uch a way that hear tree xz, yz are zero at the top and bottom face of the plate. Conequently, the expreion for u and v can be given a 1 5 z w u z z 4 3 h x 1 5 z w v z z 4 3 h y.4. Kinematic and contitutive equation (6b) Baed on the aumption made in the preceding ection, the diplacement field can be obtained uing Eq. (4)-(6) a w b 1 5 z w U( x, y, z) u( x, y) z z x 4 3h x V( x, y, z) v( x, y) z z y 4 3h y W( x, y, z) wb( x, y) w( x, y) wb 1 5 z w (7) The train aociated with the diplacement in Eq. (7) are where yz g yz xz g xz z b x x zx f x b y y zy f y b xy xy zxy f xy x x x y y y u v b wb xy, xy y x xy w xy xy w w yz, xz y x 1 5 z 5 z f z z, g h 4 h u b wb w x, x, x v b wb w y, y, y (8) (9) For elatic and iotropic FGM, the contitutive relation can be written a x Q11 Q1 x Q Q y 1 y xy Q66 xy (1) yz Q44 yz zx Q 55 zx where x, y xy, yz zx, and x, y, xy, yz, zx are the tre and train component, repectively. Uing the material propertie defined in Eq. (3), tiffne coefficient Q ij can be expreed a.5. Governing equation Ez ( ) Q11 Q 1 Ez ( ) Q1 1 Ez ( ) Q44 Q55 Q66 1 (11) The governing equation of equilibrium can be derived by uing the principle of virtual diplacement. The principle of virtual wor in the preent cae yield h/ h/ ( x x y y xy xy yz yz xzxz )ddz qwd (1) where i the top urface, and q the external load. Subtituting Eq. (7)-(8) and Eq. (1) into Eq. (1) and integrating through the thicne of the plate, Eq. (1) can be rewritten a b b b b N N N M M ( x x y y xy xy x x y y b b Mxy xy Mx x M y y Mxy xy Syz yz Sxzxz )d q( wb wb) d (13) where the tre reultant N, M b, M and S are defined by h / [ Nx Ny Nxy] [ / x y xy]dz h b b b h / [ M x M y Mxy ] [ ] d / x h y xy z z (14) h / [ M x M y Mxy ] [ ] d h / x y xy f z h / [ Sxz Syz ] [ ] d h / xz yz g z Subtituting Eq. (1) into Eq. (14) and integrating through the thicne of the plate, the tre reultant are given a N A B B b b M B D D M B D H (15) Syz A44 yz Sxz A55 xz

5 438 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) No.4 where T N [ Nx Ny Nxy] b b b b T M [ Mx M y Mxy] T M [ Mx M y Mxy] [ x y xy] b b b b [ x y xy] [ x y xy] A11 A1 A1 A A A 66 B11 B1 B1 B B B 66 D11 D1 D1 D D D 66 B11 B1 B B1 B B66 D11 D1 D D1 D D66 H11 H1 H H1 H H 66 where A ij, B ij, etc., are the plate tiffne, defined by (16a) (16b) (16c) (16d) [ Aij Bij Dij Eij Fij Hij ] 3 hn [1 z z z z z ] Qijdz hn n1 (, i j 1,,6) 1 5 Bij Bij E (, 1,,6) ij i j 4 3h 1 5 Dij Dij F (, 1,,6) ij i j 4 3h Hij Dij F (, 1,,6) ij H 4 ij i j 16 6h 9h 3 hn 1 4 [ Aij Dij Fij ] [1 z z ] Qijdz hn n1 (, i j 4, 5) A ij A (, 4,5) 4 16 ij D 6 ij F ij i j h h (17) The governing equation of equilibrium can be derived from Eq. (13) by integrating the diplacement gradient by part and etting the coefficient u, v, w b, and w to zero eparately. Thu, one can obtain the equilibrium equation aociated with the preent RPT for the FGM andwich plate N N x xy u : x y Nxy Ny v : x y b b b M M x xy M y wb : q x xy y M M x xy M y S xz w : x x y y x S yz q y (18) Subtituting Eq. (15) into Eq. (18), we obtain the following operator equation: L f (19) where u v w b w ; f =[ q q] T i a generalized force vector, and L i the ymmetric matrix of differential operator, L 11 A11d11 A66d L1 ( A1 A66 ) d1 L13 B11d111 ( B1 B66 ) d1 L14 B11d111 ( B1 B66 ) d1 L Ad A66d11 L3 Bd ( B1 B66 ) d11 L4 Bd ( B1 B66 ) d11 L33 D11d1111 ( D1 D66 ) d11 Dd L34 D11d1111 ( D1 D66 ) d11 Dd L44 H11d1111 ( H1 H66 ) d11 Hd A55d11 A44d () in which d ij, d ijl, and d ijlm are the following differential operator: 3 dij, dijl xixj xixjxl 4 dijlm (, i j, l, m1,) xixjxlxm (1)

6 No.4 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) Analytical olution for FGM andwich plate Rectangular FGM andwich plate are generally claified in accordance with the type upport ued. We are here concerned with the analytical olution of Eq. (19) for imply upported FGM andwich plate. The following boundary condition are impoed at the ide edge. v(, y) wb(, y) w(, y) wb w (, y) (, y) y y vay (, ) wb( ay, ) w( ay, ) wb w ( ay, ) ( ay, ) y y b Nx(, y) Mx(, y) Mx(, y) b Nx( a, y) Mx( a, y) Mx( a, y) ux (,) wb( x,) w( x,) wb w ( x,) ( x,) x x uxb (, ) wb ( xb, ) w ( xb, ) wb w ( xb, ) ( xb, ) x x b Ny( x,) M y( x,) M y( x,) b Ny( x, b) M y( x, b) M y( x, b) () To olve thi problem, Navier preented the external force for the cae of inuoidally ditributed load a qxy (, ) qin( x)in( y), (3) where a and =b, and q repreent the intenity of the load at the plate center. Following the Navier olution procedure, we aume the following olution form for [u v w b w ] that atifie the boundary condition, u Uco( x)in( y) v Vin( x)co( y) w b W b in( x)in( y) w W in( x)in( y) (4) where U, V, W b and W are arbitrary parameter to be determined ubject to the condition that the olution in Eq. (4) atifie the baic Eq. (19). One obtain the following operator equation K F (5) where and F denote the column and [ U V W b W ] T F [ q q] (6) a11 a1 a13 a14 a1 a a3 a 4 K (7) a13 a3 a33 a34 a14 a4 a34 a44 where a11 A11 A66 a1 ( A1 A66 ) a13 [ B11 ( B1 B66) ] a14 [ B11 ( B1 B66 ) ] a A66 A a3 [( B1 B66 ) B ] (8) a4 [( B1 B66 ) B ] 4 4 a33 D11 ( D1 D66 ) D a34 D ( D1 D66 ) D 4 a44 H11 ( H11 H66 ) 4 H A55 A44 3. Numerical Reult and Dicuion In thi tudy, tatic analyi of imply upported FGM plate by the preent RPT i uggeted for invetigation. Navier olution for bending analyi of FGM plate are preented by olving Eq. (5). The tatic analyi i conducted for two combination of metal and ceramic. The firt et of material choen i aluminum and alumina. The econd combination of material conit of aluminum and zirconia. The material propertie are a follow: Ceramic (zirconia, ZrO ): Ec 151 GPa,.3 Ceramic (alumina, AlO 3): Ec 38 GPa,.3 Metal (Aluminium, Al): Em 7 GPa,.3 Numerical reult are preented in term of non-dimenional tree and deflection. The variou nondimenional parameter ued are 1hE a b Central deflection: W W, qa 1h a b h Axial tre: x,, x qa h b Shear tre: xz xz,, qa where E =1 GPa. It i aumed (unle otherwie tated) that the plate i made from aluminium-zirconia FGM. We alo tae the hear correction factor K=5/6

7 44 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) No.4 in the firt hear deformation plate theory (FSDPT). In order to prove the validity of the preent theory, reult are obtained for fully FGM plate and compared with the exiting one in the literature [8-9]. For other verification, the reult obtained for FGM andwich plate uing the preent RPT are compared with other theorie exiting in the literature uch a CPT, FSDPT, and higher-order hear deformation plate theorie (HSDPT) uch a: TSDPT and inuoidal hear deformation plate theory (SSDPT) a hown in Table 1, in which x and y are the rotation of the normal of the midplane about y and x, repectively. Table preent the dimenionle deflection and tree of the quare FGM plate (a/h=1) for different value of the power-law. It can be een that the reult obtained uing the preent RPT are identical to thoe of Zenour [8]. Table 3 preent the central deflection and the tranvere hear tree of imply upported quare FGM plate. The reult are compared with thoe of D higher-order theory of Matunaga [9]. It can be een that the reult agree well with that reported by Matunaga [9]. Table 1 Kinematical aumption of different plate theorie Theory CPT FSDPT TSDPT SSDPT Aumption of three-dimenional diplacement w U( x, y, z) u( x, y) z x w V( x, y, z) v( x, y) z y W( x, y, z) w( x, y) U( x, y, z) u( x, y) zx ( x, y) V( x, y, z) v( x, y) z y ( x, y) W( x, y, z) w( x, y) 3 4z w U( x, y, z) u( x, y) zx( x, y) x 3h x 3 4z w V( x, y, z) v( x, y) zy( x, y) y 3h y W( x, y, z) w( x, y) w h z U( x, y, z) u( x, y) z in x ( x, y) x h w h z V( x, y, z) v( x, y) z in y ( x, y) x h W( x, y, z) w( x, y) Table Comparion of dimenionle deflection and tree in quare FGM plate (Al/Al O 3 ) ubject to inuoidal ditributed load W 3 1 he c a, b W 4 aq h a b h h b x x,, xz xz,, aq aq Ref. [8] Preent RPT Ref. [8] Preent RPT Ref. [8] Preent RPT Ceramic Metal Table 3 Comparion of dimenionle deflection and tranvere hear tree in quare FGM plate (Al/Al O 3 ) ubject to inuoidal ditributed load. a/h 5 1 W WE q h xz xz (, b/ ) / q c /( ) Ref. [9] Preent RPT Ref. [9] Preent RPT

8 No.4 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) The numerical reult of imply upported quare FGM plate of Type A are preented in Table 4-6. Thee table conider the cae of homogeneou hardcore in which the elatic modulu of Layer 1 are E c =151 Gpa (P 1, zirconia) at the top face and E m = 7 Gpa (P, aluminium) at the bottom face. The reult are conidered for =, 1,, 5, 1 and different type of andwich plate. It can be een that the reult obtained by TSDPT and the preent RPT are identical. Table 4 how that the effect of hear deformation i to increae the deflection. The difference between the hear deformation theorie i inignificant for fully ceramic plate ( = ). It i to be noted that the CPT yield identical axial tree with the FSDPT and o Table 5 lac the reult of CPT. The difference between everal ind of andwich plate i inignificant for fully ceramic plate ( = ). The axial tre i very enitive to the variation of. Table 6 how that the tranvere hear tree a per the FSDPT may be inditinguihable. The difference between everal ind of andwich plate i inignificant for fully ceramic plate ( = ). The value of the tranvere hear tree decreae a decreae for the FGM plate with homogeneou hardcore. Table 4 Comparion of dimenionle deflection of imply upported quare power-law FGM plate with other theorie (a/h=1) Theory W Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT Table 5 Comparion of dimenionle axial tre of imply upported quare power-law FGM plate with other theorie (a/h=5) Theory Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT x

9 44 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) No.4 In general, the fully ceramic plate give the mallet deflection and hear tree and the larget axial tree. A the volume fraction exponent increae for FGM plate, the deflection, axial tre and hear tre will increae. Thee reult will decreae a the core thicne, with repect to the total thicne of the plate, increae. In fact the non-ymmetric --1 FGM plate yield the mallet axial tree. The comparion of the maximum deflection are given in Table 7 for FGM andwich plate with homogeneou hardcore and with volume fraction indice =. It can be oberved that the reult obtained by the preent RPT are identical to thoe of TSDPT. In addition, the deflection will decreae a the apect ratio a/b increae. Table 6 Comparion of dimenionle tranvere hear tre of imply upported quare power-law FGM plate with other theorie (a/h=5) Theory Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT Preent RPT SSDPT TSDPT FSDPT xz Table 7 Effect of apect ratio a/b on the dimenionle deflection of FGM andwich plate (a/h=1 and =). Scheme Theory W a/b=1/3 a/b=.5 a/b=1 a/b=1.5 a/b= Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT Preent RPT SSDPT TSDPT FSDPT CPT

10 No.4 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) The central deflection due to flexure w b and hear w are calculated, and then are ummed up to get the total deflection W. Thee reult are hown in Table 8 for quare power-law FGM plate of Type A with homogeneou oftcore in which the elatic modulu of Layer 1 i E m =7 GPa (P 1, aluminum) at the top face a/h and E c =151 GPa (P, zirconia) at the bottom face. The error level increae with the decreae in the thicne ratio a/h. Hence, the hear effect i more pronounced for maller thicne ratio a/h. The reult emphaize alo the great influence exerted by the volume fraction indice on the analyzed deflection. Table 8 Dimenionle deflection of quare FGM andwich plate with hear effect Fig. 3-8 preent ome numerical reult of imply upported quare power-law FGM plate of Type A uing the preent RPT. Fig. 3-5 conider the cae of homogeneou hardcore in which the elatic modulu of Layer 1 i E c =151 GPa (P 1, zirconia) at the top face and E m =7 GPa (P, aluminum) at the bottom face. Fig. 6-8 conider the cae of homogeneou oftcore in which the elatic modulu of Layer 1 i E m =7 GPa (P 1, aluminum) at the top face and E c =151 GPa (P, zirconia) at the bottom face. Fig.3 how the variation of the center deflection -1- w wb W Error/% Fig. 3 Dimenionle center deflection W a a function of ide-to-thicne ratio a/h of FGM andwich plate (Type A) with homogeneou hardcore for variou value of and different type of andwich plate. with ide-to-thicne ratio a/h for different type of FGM ymmetric plate with a homogeneou hardcore. The deflection of the metallic plate i found to be the larget magnitude and that of the ceramic plate of the mallet magnitude. The deflection of the FGM andwich plate increae when a/h5. All the plate

11 444 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) No.4 with intermediate propertie undergo correponding intermediate value of center deflection. Thi i expected becaue the metallic plate i the one with the lowet tiffne and the ceramic plate i the one with the highet tiffne. Fig. 4 contain the plot of the axial tre x through the thicne of the plate with a homogeneou hardcore and for =, 1,. Under the application of the inuoidal loading, the tree are tenile at the top urface and compreive at the bottom urface. The homogeneou ceramic plate ( = ) yield the maximum compreive (tenile) tre at the bottom (top) urface. Thee are the metal-rich urface for the FGM plate ( = 1, ). Note that for the different volume fraction exponent choen, the plate correponding to = yield the maximum compreive (tenile) tre at the bottom (top) urface of the core layer (ee Fig. 4(a)-4(c). Thee are the ceramic-rich urface in Fig. 4 Variation of normal tre x through plate thicne of FGM andwich plate (Type A) with homogeneou hardcore for variou value of and different type of andwich plate (a/h=1). which the ceramic plate experience the minimum compreive or tenile tree. The tre profile for plate made of pure material (ceramic) change linearly through the thicne. However, the axial tre variation i not linear for FGM plate. In Fig. 5 we have plotted the tranvere hear tre xz through the thicne of the plate with a homogeneou hardcore and for =, 1,. The maximum value occur at a point on the mid-plane of the plate and it magnitude for FGM plate i larger than that for homogeneou ceramic plate ( = ).

12 No.4 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) Fig. 5 Variation of tranvere hear tre xz through plate thicne of FGM andwich plate (Type A) with homogeneou hardcore for variou value of and different type of andwich plate (a/h=1). Fig. 6 how the variation of the center deflection with ide-to-thicne ratio for different type of FGM ymmetric plate with a homogeneou oftcore in which the elatic modulu of Layer 1 i E m =7 GPa (P 1, aluminum) at the top urface and E c =151 GPa (P, zirconia) at the bottom urface. Contrary to the cae of homogeneou hardcore, it can be oberved that for FGM plate with a homogeneou oftcore, tranvere deflection decreae a power law exponent i increaed. All the plate with intermediate propertie undergo correponding intermediate value of center Fig. 6 Dimenionle center deflection W a a function of ide-to-thicne ratio a/h of FGM andwich plate (Type A) with homogeneou oftcore for variou value of and different type of andwich plate. deflection. Thi i expected becaue the metallic plate i the one with the lowet tiffne and the ceramic plate i the one with the highet tiffne. Fig. 7 depict the through-the-thicne ditribution of the axial tre x in the FGM andwich plate with a homogeneou oftcore and for =, 1,. The tree are tenile at the top urface and compreive at the bottom urface. The homogeneou metal plate ( = ) yield the maximum compreive (tenile) tre at the bottom (top) urface. The tre profile for plate made of pure material (metal) change linearly through the thicne. However, the axial tre variation i not linear for FGM plate.

13 446 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) No.4 The plot of hear tre xz acro the FGM andwich plate with a homogeneou oftcore i preented in Fig. 8. The maximum value occur at a point on the mid-plane of the plate and it magnitude for homogeneou metal plate ( = ) i larger than that for FGM plate. Uing the preent RPT, we preent in Fig the center deflection W, axial tre x and tranvere hear tre xz for the andwich plate of Type B (FGM core) with = 1,, 5. P 1 i referred to the propertie of aluminum and P the propertie of zirconia. In thi cae, the FGM core i metal-rich at the top urface and ceramic-rich at the bottom urface. Fig. 9 how the variation of the center deflection for variou power law exponent and with the ide-tothicne ratio. The FGM plate deflection i between thoe of plate made of ceramic and metal. Fig. 7 Variation of normal tre x through plate thicne of FGM andwich plate (Type A) with homogeneou oftcore for variou value of and different type of andwich plate (a/h=1).

14 No.4 Hadj Henni ABDELAZIZ et al. / Chinee Journal of Aeronautic 4(11) andwich plate. Contrary to the plate of Type A, the ditribution of the hear tre i not ymmetric in the cae of andwich plate of Type B (FGM core). Fig. 8 Variation of tranvere hear tre xz through plate thicne of FGM andwich plate (Type A) with homogeneou oftcore for variou value of and different type of andwich plate (a/h=1). Fig. 11 Dimenionle tranvere hear tree xz of the andwich quare plate with FGM core (a/h= 1). 4. Concluion Fig. 9 Dimenionle center deflection W of the andwich quare plate with FGM core. A exhibited in Fig. 1, the axial tre, x, i compreive throughout the plate up to almot z = z/h=.6 and then they become tenile. The maximum compreive tree occur at a point on the bottom urface and the maximum tenile tree occur, of coure, at a point on the top urface of the FGM plate. A two variable refined theory i extended to the bending repone of rectangular FGM andwich plate. The number of primary variable in thi theory i even le than that of firt- and higher-order hear deformation plate theorie. Hence, unlie any other theory, the theory preented give rie to only four governing equation reulting in coniderably lower computational effort when compared with the other higherorder theorie reported in the literature having more number of governing equation. The theory account for parabolic ditribution of the tranvere hear train through the plate thicne, and atifie the zero traction boundary condition on the urface of the plate without uing hear correction factor. The accuracy and efficiency of the preent theory have been demontrated for bending behavior of imply upported FGM andwich plate. The power-law FGM andwich plate with FGM faceheet and homogeneou core, and the andwich plate with homogeneou faceheet and FGM core are conidered. The center deflection, axial tre and tranvere hear tre predicted by the preent theory (with four unnown) and TSDPT (five unnown) are identical. In concluion, it can be aid that the propoed theory i not only accurate but alo efficient in predicting the deflection and tree of FGM andwich plate. Reference Fig. 1 Dimenionle axial tree x of the andwich quare plate with FGM core (a/h=1). Fig. 11 depict the through-the-thicne ditribution of the hear tree xz in the quare FGM [1] Yamanouchi M, Koizumi M, Shiota I. Functionally gradient material. Proceeding of the Firt International Sympoium on Functionally Gradient Material [] Fuui Y. Fundamental invetigation of functionally gradient material manufacturing ytem uing centrifugal force. Japan Society Mechanical Engineer International Journal 1991; 34(1): [3] Koizumi M. FGM activitie in Japan. Compoite Part

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