A quasi-3d hyperbolic shear deformation theory for functionally graded plates

Transcription

1 A quai-3d hyperolic hear deformation theory for functionally graded plate Huu-Tai Thai a,*, Thuc P. Vo, Tinh Q. Bui c, Trung-Kien Nguyen d a School of Civil and Environmental Engineering, The Univerity of New South Wale, Sydney, Autralia Faculty of Engineering and Environment, Northumria Univerity, Newcatle upon Tyne, NE1 8ST, UK. c Department of Civil Engineering, Univerity of Siegen, Paul-Bonat-Str. 9-11, 5776, Siegen, Germany. d Faculty of Civil Engineering and Applied Mechanic, Univerity of Technical Education Ho Chi Minh City, 1 Vo Van Ngan Street, Thu Duc Ditrict, Ho Chi Minh City, Vietnam Atract A quai-3d hyperolic hear deformation theory for functionally graded plate i developed. The theory account for oth hear deformation and thickne tretching effect y a hyperolic variation of all diplacement acro the thickne, and atifie the tre-free oundary condition on the top and ottom urface of the plate without requiring any hear correction factor. The enefit of the preent theory i that it contain le numer of unknown and governing equation than the exiting quai-3d theorie, ut it olution are compared well with 3D and quai-3d olution. Equation of motion are derived from Hamilton principle. Analytical olution for ending and free viration prolem are otained for imply upported plate. Numerical example are preented to verify the accuracy of the preent theory. Keyword: functionally graded plate; higher-order theory; ending; viration * Correponding author. Tel.: addre: thaihuutai@hanyang.ac.kr (H.T. Thai). 1

2 1. Introduction Functionally graded material (FGM) are a type of nonhomogeneou compoite material, in which the material propertie vary moothly and continuouly from one urface to another. A typical FGM i made from a mixture of two material phae, for example, a ceramic and a metal. An advantage of FGM over laminated compoite i that it eliminate the delamination mode of failure found in the laminated compoite. In addition, the material propertie of FGM can e tailored to different application and working environment. Thi make FGM preferale in many tructural application uch a nuclear reactor, aeropace, mechanical, automotive, and civil engineering. Since the hear deformation effect are more pronounced in advanced compoite like FGM, hear deformation theorie uch a firt-order hear deformation theory (FSDT) and higher-order hear deformation theorie (HSDT) hould e ued. The FSDT [1-9] give acceptale prediction, ut require a hear correction factor which i hard to find out conitently ecaue of dependent on many parameter including geometry, oundary condition, and loading condition. The HSDT [1-17] do not require a hear correction factor, ut their equation of motion are more complicated than thoe of the FSDT. It hould e noted that the thickne tretching effect (i.e., e = ) i ignored in oth the FSDT and HSDT y auming a contant tranvere diplacement through the thickne of the plate. Although thi aumption i appropriate for moderately thick functionally graded (FG) plate, ut i inaccurate for thick FG one [18]. The importance of the thickne tretching effect in FG plate ha een pointed out in the work of Carrera et al. [19]. Quai-3D theorie are HSDT in which the tranvere diplacement i expanded a a higher-order variation through the thickne of the plate, and hence, thickne tretching

3 effect i captured. There are everal quai-3d theorie propoed in the literature. For example, Kant and Swaminathan [] propoed a quai-3d theory with all diplacement component expanded a a cuic variation through the thickne. The theorie preented y Chen et al. [1], Talha and Singh [], Reddy [3], and Neve et al. [4] are aed on a cuic variation of in-plane diplacement and a quadratic variation of tranvere diplacement. Intead of uing polynomial function, Ferreira et al. [5] employed the inuoidal function for all diplacement component. Neve et al. [6-7] employed the polynomial and the non-polynomial (inuoidal [6] and hyperolic [7]) function for tranvere and in-plane diplacement, repectively. It hould e noted that the aovementioned quai-3d theorie are too cumerome and computationally expenive ince they handle many unknown (e.g., theorie y Ref. [] with twelve unknown, Ref. [1-3] with eleven unknown, and Ref. [5-7,4] with nine unknown). Recently, Mantari and Guede Soare [8] preented a generalied formulation in which many hyrid quai-3d theorie with ix unknown can e derived. Although the hyrid quai- 3D theorie [8] have ix unknown, they are till more complicated than the FSDT. A a conequence, a imple quai-3d theory propoed in the preent work i neceary. Thi work aim to develop a imple quai-3d theory with only five unknown for ending and free viration analyi of FG plate. The diplacement field i choen aed on a generalied formulation [8] with a hyperolic variation for all diplacement. By dividing the tranvere diplacement into the ending and hear part, the numer of unknown of the theory i reduced, and thu aving computational time. Equation of motion derived from Hamilton principle are analytically olved for ending and free viration prolem of a imply upported plate. Numerical example are preented to verify the accuracy of the preent theory. 3

4 . Theoretical formulation A mentioned aove, the diplacement field of the preent theory i choen aed on the generalied formulation with a hyperolic variation for all diplacement component. In fact, the ue of hyperolic function wa firt propoed y Soldato [9], later ued y Xiang et al. [3], Akavci [31], and El Meiche et al. [3], and recently y Neve et al. [7]. According to Ref. [33,8], the diplacement field take the form w u1 ( x, y,, t) = u ( x, y, t) - + Y ( ) jx x w u ( x, y,, t) = v( x, y, t) - + Y ( ) j y y 3 (,,, ) = (,, ) + Y ( ) j (,, ) u x y t w x y t x y t (1) where u, v, w, j x, j y and j are ix unknown diplacement function of midplane of the plate; and Y ( ) i a hape function repreenting the ditriution of the tranvee hear train and hear tree through the thickne. In thi tudy, the hape function i choen aed on the hyperolic function propoed y Soldato [9] a æ ö æ 1 ö Y ( ) = hinhç - cohç è h ø è ø () with h eing the thickne of the plate. By deviding the tranvere diplacement w into ending and hear part (i.e., w = w + w ) and making further aumption given y j = w / x and j = w / y, the diplacement field of the preent theory can e x rewritten in impler form a y w w u1 ( x, y,, t) = u ( x, y, t) - - f ( ) x x w w u ( x, y,, t) = v( x, y, t) - - f ( ) y y 3 (,,, ) = (,, ) + (,, ) + ( ) j (,, ) u x y t w x y t w x y t g x y t (3) 4

5 where f ( ) = - Y ( ) and g ( ) ( ) 1 f ( ) coh ( / h) coh( 1/ ) = Y = - = -. Clearly een that the diplacement field in Eq. (3) handle only five unknown, i.e., u, v, w, w, j. The train aociated with the diplacement field in Eq. (3) are: u w w e x = - - f x x x ( ) (4a) e y v w w = - - f y y y ( ) (4) ( ) e = g j (4c) g g g xy x y u v w w = f ( ) y x x y x y æ w j ö = g ( ) ç + è x x ø æ w j ö = g ( ) ç + è y y ø (4d) (4e) (4f) It can e een from Eq. (4e) and (4f) that the tranvere hear train ( g x, g y ) are equal to ero at the top ( = h / ) and ottom ( = - h / ) urface of the plate. A hear correction factor i, therefore, not required. The contitutive relation of a FG plate can e written a ì x ü éc11 C1 C13 ù ìe x ü ê y C1 C C3 ú e y ê ú êc13 C3 C33 ú e í ý = ê ú í ý xy ê C66 ú g xy ê x C55 ú g x ê ú î y þ êë C44 úû î g y þ (5) where C ij are the three-dimenional elatic contant determined y 5

6 C = C = C = C = C = C = C = C = C = ( 1-n ) E ( 1- n )( 1+ n ) n E ( 1- n )( 1+ n ) E 1 ( + n ) (6a) (6) (6c) with E and n eing Young modulu and Poion ratio, repectively, of a FG plate. Hamilton principle i ued herein to derive the equation of motion. The principle can e tated in analytical form a T ( d d d ) ò U + V - K dt = (7) where d U i the variation of train energy; d V i the variation of work done y external force; and d K i the variation of kinetic energy. The variation of train energy i given explicitly y h / ò ò A - h / ( x x y y xy xy x x y y ) du = de + de + de + dg + dg + dg x x x y y y + Qx + dad é du d w d w d v d w d w = ò N M P N M P A ê ë x x x y y y æ du d v ö d w d w + Rdj + N xy ç + - M xy - Pxy è y x ø x y x y æ d w dj ö æ d w dj öù ç + Qy ç + ú da è x x ø è y y øû (8) where N, M, P, Q, and R are the tre reultant defined y h / ( x, y, xy ) ( x, y, xy ) N N N = ò d (9a) -h / h / ( x, y, xy ) ( x, y, xy ) M M M = ò d (9) -h / 6

7 h / ( x, y, xy ) ( x, y, xy ) ( ) P P P = ò f d (9c) h / -h / ( Qx, Qy ) ( x, y ) ( ) h / -h / = ò g d (9d) -h / ( ) R = ò g d (9e) Sutituting Eq. (4) into Eq. (5) and the uequent reult into Eq. (9), the tre reultant can e expreed in term of generalied diplacement (,,,, ) u v w w j a ì u ü x v ù 13 y ú u v ú y x ú w - ú x ú w ú - 3 y ú í ý w ú - x x 11 B1 D11 D1 H11 H1 Y ú 13 ú w x 3 ú ú w B66 D66 H - 66 ú y ú 33 û w - x y j î þ ì N x ü é A A B B B B X N ê y ê A A B B B B X N ê xy A B B ê M x ê B B D D D D Y M y ê B B D D D D Y í ý = ê M xy ê B D D P ê x B ê Py ê B B D D H H Y P ê xy ê î R þ êë X X Y Y Y Y Z (1a) w j ìqx ü éa55 ù ì ü + x x í ý = w Q ê ú í j y A ý î þ ë 44 û + î y y þ (1) where h / (,,,,,, ) ( 1,,,,,, ) A A B B D D H = ò g f f f C d (11a) ij ij ij ij ij ij ij ij -h / h / (,,, ) ( 1,,, ) X Y Y Z = ò f g g C d (11) ij ij ij ij ij -h / The variation of work done y externally tranvere load q can e expreed a 7

8 The variation of kinetic energy i ( ) ò A (1) dv = - qd w + w + gj da h / ò ò A - h / A ( & & & & & & ) d K = r u du + u du + u du dad ò { I u& du& v& d v& ( w& w& ) d ( w& w& ) J ( w& w& ) dj& & jd ( w& w& ) = éë ùû + éë ùû æ d w& w& d w& w& ö æ w& d w& w& d w& ö - I1 ç u& + du& + v& + d v& + I ç + è x x y y ø è x x y y ø æ d w& w& d w& w& ö æ w& d w& w& d w& ö - J1 ç u& + du& + v& + d v& + K ç + è x x y y ø è x x y y ø æ w& d w& w& d w& w& d w& w& d w& ü ö + J ç K & j dj& ýda è x x x x y y y y ø þ (13) where dot-upercript convention indicate the differentiation with repect to the time variale t ; r i the ma denity; and (,, ) defined y h / I J K are the ma moment of inertia i i i (, 1, ) ( 1,, ) I I I = ò rd (14a) -h / h / (,, ) (,, ) J J J = ò g f f rd (14) 1 -h / h / (, ) (, ) K K = ò g f rd (14c) -h / The equation of motion can e otained y utituting the expreion for d U, d V, and d K from Eq. (8), (1), and (13) into Eq. (7), integrating y part, and collecting the coefficient of d u, d v, d w, d w, and dj. N N x xy w&& w&& du : + = Iu&& - I1 - J1 (15a) x y x x N xy N y w&& w&& d v : + = Iv&& - I1 - J1 (15) x y y y 8

9 M M xy M x y d w : q x x y y æ u&& v&& ö = I ( w&& + w&& ) + J && j + I ç + - I Ñ w&& - J Ñ w&& è x y ø 1 P Pxy P x y Q Q x y d w : q x x y y x y æ u&& v&& ö = I ( w&& + w&& ) + J && j + J ç + - J Ñ w&& - K Ñ w&& è x y ø 1 (15c) (15d) Q Q x y dj : + - R + gq = J ( w&& + w&& ) + K && j (15e) x y Sutituting Eq. (1) into Eq. (15), the equation of motion of the preent theory can e expreed in term of diplacement (,,,, ) u v w w j a u u v w w A A A A B B B x y x y x x y ( ) ( ) w w j w&& w&& -B - B + B + X = I u - I - J x x y x x x ( 1 66 ) && v v u w w A + A 66 + ( A1 + A66 ) - B - 3 ( B1 + B66 ) y x x y y x y w w j w&& w&& -B - B + B + X = I v - I - J y x y y y y ( 1 66 ) && u æ u v ö v w w B B B B D D x è x y x y ø y x y ( ) ç w w w w - ( D1 + D66 ) - D11 - D - ( D1 + D66 ) x y x y x y j j æ u&& v&& ö + Y13 + Y 3 + q = I ( w&& + w&& ) + J && j + I1 ç + - I Ñ w&& - J Ñ w&& x y è x y ø (16a) (16) (16c) 9

10 u æ u v ö v w w B B B B D D x è x y x y ø y x y ( ) ç w w w w - ( D1 + D66 ) - H11 - H - ( H1 + H66 ) x y x y x y w w j j + A55 + A 44 + ( Y13 + A55 ) + ( Y3 + A44 ) + q x y x y æ u&& v&& ö = I ( w&& + w&& ) + J && j + J ç + - J Ñ w&& - K Ñ w&& è x y ø 1 u v w w w w -X - X + Y + Y + Y + A + Y + A x y x y x y ( ) ( 3 44 ) j j + A + A - Z + gq = J w + w + K x y j ( && && ) && j (16d) (16e) 3. Analytical olution Conider a imply upported rectangular plate with length a and width under tranvere load q. Baed on Navier olution method, the following expanion of diplacement (,,,, ) u v w w j are aumed a ( ) iwt u x, y, t = U e coa xin y ( ) ( ) ( ) ( ) m= 1 n= 1 iwt v x, y, t = V e ina x co y m= 1 n= 1 iwt w x, y, t = W e ina xin y m= 1 n= 1 iwt w x, y, t = W e ina xin y m= 1 n= 1 iwt j x, y, t = f e ina xin y åå åå åå åå åå m= 1 n= 1 where i = - 1, a = mp / a, np / mn mn mn mn mn =, (,,,, ) mn mn mn mn mn (17) U V W W f are the unknown maximum diplacement coefficient, and w i the viration frequency. The tranvere load q i alo expanded a 1

11 ( ) q x, y = åå Q ina xin y (18) m= 1 n= 1 mn For the cae of inuoidal load, coefficient Qmn = q repreent the intenity of the load at the plate center. Sutituting Eq. (17) and (18) into Eq. (16), the analytical olution can e otained y æ ék11 k1 k13 k14 k15 ù ém11 m13 m14 ù ö ìu mn ü ì ü ç ê k1 k k3 k4 k ú ê 5 m m3 m4 ú V ç ê ú ê ú mn ç k13 k3 k33 k34 k35 w m13 m3 m33 m34 m ê ú - ê ú 35 íwmn ý = íqmn ý ç ê ú ê ú ç êk14 k4 k34 k44 k45 ú êm14 m4 m34 m44 m 45 ú W mn Q mn ç êk15 k5 k35 k45 k ú ê 55 m35 m45 m ú è ë û ë 55 û ø î fmn þ î þ (19) where ( ) k = A a + A, k = A a + A, k = A + A a, ( ), ( ), ( ), ( ), ( ) ( A55 ) a + ( Y3 + A44 ) ( ) k = -B a - B + B a k = -B a - B + B a k = -X a k = -B - B + B a k = -B - B + B a k = -X k = D a + D + D a + D, k = Y k = D a + D + D a + D, k = A a + A + Z ( ) = + ( a + ), = ( a ) = + ( a + ), =, = k = Y a + Y, k = H a + H + H a + H + A a + A m = I, m = - a I, m = - a J, m = I, m = - I, m = - J m I I m m I K m J m K I + J +, m = J () 4. Numerical reult 4.1. Reult for ending analyi Conider a imply upported FG plate ujected to inuoidal load. The effective Young modulu E ( ) i aumed to vary exponentially through the thickne of the plate a [33] p(.5 + / h) ( ) ( ) ( ) E = E f f = e (1), 11

12 p where E = E and Et = Ee denote Young modulu of the ottom and top urface of the FG plate, repectively; E i Young modulu of the homogeneou plate; and p i a parameter that indicate the material variation through the thickne and take value greater than or equal to ero. The variation of the exponential function f ( ) through the thickne of the plate i preented in Fig. 1 for different value of p. Poion ratio i aumed to e contant n =.3. For convenience, the following dimenionle form are ued: 1E h æ ö 1E h æ a ö u ( ) = u ç,,, w( ) = wç,, q a è ø è ø qa h æ a ö 1h ç ( ) =,,, ( ) = (,, ) x, y x, y xy xy qa è ø qa h æ ö h æ a ö ( ) = ç,,, ( ) = ç,, è ø è ø x x y y qa qa () The dimenionle diplacement and tre are preented in Tale 1-4 for variou value of apect ratio / a, thickne ratio a / h, and material parameter p. The through thickne variation of the dimenionle diplacement and tree are alo given in Fig. for a thick FG plate with a / h = 4 and p =.5. The otained reult are compared with the exact 3D [33] and quai-3d olution [8,33-34]. It hould e noted that the quai-3d olution [33-34] are derived aed on a trigonometric variation of oth in-plane and tranvere diplacement, while the quai-3d olution [8] are computed aed on a cuic variation of in-plane diplacement and a paraolic variation of tranvere diplacement acro the thickne. In addition, the reult of HSDT [35] are alo provided to how the importance of including the thickne tretching effect. The HSDT olution [35] i computed aed on a trigonometric variation of in-plane diplacement and a contant tranvere diplacement acro the thickne (i.e., 1

13 thickne tretching effect i omitted, e = ). It can e oerved that the otained reult are in excellent agreement with 3D and quai-3d olution, particularly with thoe reported y Mantari and Guede Soare [8,34]. The preent quai-3d theory i even more accurate than the quai-3d inuoidal theory [33]. Since the preent quai-3d theory and other quai-3d theorie include the thickne tretching effect, their olution are very cloe to each other. Meanwhile, the HSDT [35], which omit thi effect, give inaccurate prediction and lightly overetimate the deflection epecially for very thick plate (i.e., a / h =, ee Tale 1 and 3). The error in the HSDT are reduced with increaing the thickne ratio a / h. In general, the preent quai-3d theory i highly accurate and comparale to 3D olution even in the cae of very thick plate, e.g., a / h =. It i worth noting that the developed theory conit of five unknown, while the numer of unknown in the HSDT [35] and other quai-3d theorie [8,33-34] i five and ix, repectively. Conequently, it may e concluded that the preent quai-3d theory i not only more accurate than the HSDT having the ame five unknown, ut alo comparale with the quai-3d theorie having more numer of unknown. 4.. Reult for free viration analyi The accuracy of the propoed quai-3d theory i alo verified through the free viration analyi. Conider a imply upported Al/ZrO plate made from a mixture of a metal (Al) and a ceramic (ZrO ). Young modulu and denity of the metal are E = 7 GPa and r = 7 kg/m 3, repectively, and that of ceramic are E = 38 GPa m m and r c = 38 kg/m 3, repectively. Poion ratio i aumed to e contant and equal to.3. The effective Young modulu i etimated uing the power-law ditriution with Mori-Tanaka cheme. According to the power-law ditriution with Mori-Tanaka c 13

14 cheme, the ulk modulu K ( ) i given y [36] ( ) = + ( - ) K K K K V c m c m K c -K m 1+ Vm K m + 4/3G m (3) where ucript m and c repreent the metal and ceramic contituent, repectively; G i the hear modulu; and the volume fraction of the metal phae V m and ceramic phae V c are given y V m p = 1- V and V = (.5 + / h) (4) c c with p eing the power law index. The variation of the volume fraction V c through the thickne of the plate i given in Fig. 3 for variou value of the power law index p. Recall that the ulk modulu and the hear modulu are related to Young modulu E and Poion ratio n y K = E / 3( 1- n ) and G E / ( 1 n ) = +. Thu, y rewriting Eq. (3) in term of E and n, the effective Young modulu E ( ) i rewritten y ( ) = + ( - ) E E E E m c m 1+ V V c æ E ö ç c 1 n m 1 + ç - Em 3-3 n è ø The effective denity r ( ) i etimated uing the power-law ditriution with Voigt' rule of mixture a [1] ( ) ( ) m c m c (5) r = r + r - r V (6) Tale 5 contain the dimenionle fundamental frequency w of quare plate for different value of thickne ratio and power law index. The dimenionle frequency i defined y w = wh r / E. The calculated frequencie are compared with 3D m m olution of Vel and Batra [37], quai-3d olution of Neve et al. [7,6,4], and thirdorder hear deformation (TSDT) olution of Ferreira et al. [1]. It hould e noted that 14

15 the quai-3d olution are derived aed on the inuoidal [6], hyperolic [7], and cuic [4] variation of the in-plane diplacement, and a quadratic variation of the tranvere diplacement acro the thickne. Since the propoed and quai-3d theorie include the thickne tretching effect, they lead to olution cloe to each other, and their olution match well with 3D olution [37]. Wherea, the TSDT olution [1] lightly underetimate frequency due to ignoring the thickne tretching effect. Again, it houd e noted that the numer of unknown of the propoed theory i only five a againt nine in the cae of the quai-3d theorie of Neve et al. [7,6,4]. 5. Concluion A quai-3d hyperolic hear deformation theory i developed for ending and viration analyi of FG plate. The approach contain five unknown, ut account for oth hear deformation and thickne tretching effect without the need for any hear correction factor. Equation of motion derived from Hamilton principle are analytically olved for ending and free viration prolem of a imply upported plate. By dividing the tranvere diplacement into the ending and hear part, the numer of unknown of the theory i reduced, and the computational time i thu aved. The following main point may e drawn from the preent tudy: (1) The reult predicted y the propoed theory are in an excellent agreement with 3D olution even for the cae of very thick plate with a / h =. () The preent quai-3d theory ha five unknown, ut give reult comparale with thoe predicted y the exiting quai-3d theorie having more numer of unknown. (3) The propoed theory i even more accurate than the quai-3d inuoidal theory when compared to 3D olution. (4) The thickne tretching effect i more pronounced for thick plate and it need to 15

16 e taken in conideration in the modeling. Acknowledgement The fourth author gratefully acknowledge financial upport from Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No , and from Univerity of Technical Education Ho Chi Minh City. Reference 1. Nguyen, T.K., Sa, K., Bonnet, G.: Firt-order hear deformation plate model for functionally graded material. Compo. Struct. 83(1), 5-36 (8).. Zhao, X., Lee, Y.Y., Liew, K.M.: Free viration analyi of functionally graded plate uing the element-free kp-rit method. J. Sound Vi. 319(3-5), (9). 3. Hoeini-Hahemi, S., Rokni Damavandi Taher, H., Akhavan, H., Omidi, M.: Free viration of functionally graded rectangular plate uing firt-order hear deformation plate theory. Appl. Math. Modell. 34(5), (1). 4. Hoeini-Hahemi, S., Fadaee, M., Atahipour, S.R.: A new exact analytical approach for free viration of Reiner-Mindlin functionally graded rectangular plate. Int. J. Mech. Sci. 53(1), 11- (11). 5. Irchik, H.: On viration of layered eam and plate. J. Appl. Math. Mech. 73(4-5), (1993). 6. Noier, A., Fallah, F.: Reformulation of Mindlin Reiner governing equation of functionally graded circular plate. Acta Mech. 198(3-4), 9-33 (8). 7. Saidi, A.R., Baferani, A.H., Jomehadeh, E.: Benchmark olution for free viration of functionally graded moderately thick annular ector plate. Acta Mech. 19(3-4), (11). 16

17 8. Yang, B., Ding, H.J., Chen, W.Q.: Elaticity olution for a uniformly loaded rectangular plate of functionally graded material with two oppoite edge imply upported. Acta Mech. 7(3-4), (9). 9. Zenkour, A.M., Allam, M.N.M., Shaker, M.O., Radwan, A.F.: On the imple and mixed firt-order theorie for plate reting on elatic foundation. Acta Mech. (1-4), (11). 1. Reddy, J.N.: Analyi of functionally graded plate. Int. J. Numer. Method Eng. 47(1-3), (). 11. Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.F., Martin, P.A.L.S.: Static analyi of functionally graded plate uing third-order hear deformation theory and a mehle method. Compo. Struct. 69(4), (5). 1. Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.F., Jorge, R.M.N.: Natural frequencie of functionally graded plate y a mehle method. Compoite Structure 75(1-4), (6). 13. Zenkour, A.M.: Generalied hear deformation theory for ending analyi of functionally graded plate. Appl. Math. Modell. 3(1), (6). 14. Pradyumna, S., Bandyopadhyay, J.N.: Free viration analyi of functionally graded curved panel uing a higher-order finite element formulation. J. Sound Vi. 318(1- ), (8). 15. Bodaghi, M., Saidi, A.R.: Levy-type olution for uckling analyi of thick functionally graded rectangular plate aed on the higher-order hear deformation plate theory. Applied Mathematical Modelling 34(11), (1). 16. Hoeini-Hahemi, S., Fadaee, M., Atahipour, S.R.: Study on the free viration of thick functionally graded rectangular plate according to a new exact cloed-form 17

18 procedure. Compo. Struct. 93(), (11). 17. Xiang, S., Jin, Y.X., Bi, Z.Y., Jiang, S.X., Yang, M.S.: A n-order hear deformation theory for free viration of functionally graded and compoite andwich plate. Compo. Struct. 93(11), (11). 18. Qian, L.F., Batra, R.C., Chen, L.M.: Static and dynamic deformation of thick functionally graded elatic plate y uing higher-order hear and normal deformale plate theory and mehle local Petrov Galerkin method. Compoite Part B 35(6 8), (4). 19. Carrera, E., Brichetto, S., Cinefra, M., Soave, M.: Effect of thickne tretching in functionally graded plate and hell. Compoite Part B 4(), (11).. Kant, T., Swaminathan, K.: Analytical olution for the tatic analyi of laminated compoite and andwich plate aed on a higher order refined theory. Compo. Struct. 56(4), (). 1. Chen, C.S., Hu, C.Y., Tou, G.J.: Viration and taility of functionally graded plate aed on a higher-order deformation theory. J. Reinf. Plat. Compo. 8(1), (9).. Talha, M., Singh, B.N.: Static repone and free viration analyi of FGM plate uing higher order hear deformation theory. Appl. Math. Modell. 34(1), (1). 3. Reddy, J.N.: A general nonlinear third-order theory of functionally graded plate. Int. J. Aerop. Lightweight Struct. 1(1), 1-1 (11). 4. Neve, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N., Soare, C.M.M.: Static, free viration and uckling analyi of iotropic and andwich functionally graded plate uing a quai-3d higher-order hear 18

19 deformation theory and a mehle technique. Compoite Part B 44(1), (13). 5. Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Polit, O.: Analyi of laminated hell y a inuoidal hear deformation theory and radial ai function collocation, accounting for through-the-thickne deformation. Compoite Part B 4(5), (11). 6. Neve, A.M.A., Ferreira, A.J.M., Carrera, E., Roque, C.M.C., Cinefra, M., Jorge, R.M.N., Soare, C.M.M.: A quai-3d inuoidal hear deformation theory for the tatic and free viration analyi of functionally graded plate. Compoite Part B 43(), (1). 7. Neve, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N., Soare, C.M.M.: A quai-3d hyperolic hear deformation theory for the tatic and free viration analyi of functionally graded plate. Compo. Struct. 94(5), (1). 8. Mantari, J.L., Guede Soare, C.: Generalied hyrid quai-3d hear deformation theory for the tatic analyi of advanced compoite plate. Compo. Struct. 94(8), (1). 9. Soldato, K.P.: A tranvere hear deformation theory for homogeneou monoclinic plate. Acta Mech. 94(3), 195- (199). 3. Xiang, S., Wang, K.M., Ai, Y.T., Sha, Y.D., Shi, H.: Analyi of iotropic, andwich and laminated plate y a mehle method and variou hear deformation theorie. Compo. Struct. 91(1), (9). 31. Akavci, S.: Two new hyperolic hear diplacement model for orthotropic laminated compoite plate. Mech. Compo. Mater. 46(), 15-6 (1). 19

20 3. El Meiche, N., Touni, A., Ziane, N., Mecha, I., Adda.Bedia, E.A.: A new hyperolic hear deformation theory for uckling and viration of functionally graded andwich plate. Int. J. Mech. Sci. 53(4), (11). 33. Zenkour, A.: Benchmark trigonometric and 3-D elaticity olution for an exponentially graded thick rectangular plate. Arch. Appl. Mech. 77(4), (7). 34. Mantari, J.L., Guede Soare, C.: A novel higher-order hear deformation theory with tretching effect for functionally graded plate. Compoite Part B 45(1), (13). 35. Mantari, J.L., Guede Soare, C.: Bending analyi of thick exponentially graded plate uing a new trigonometric higher order hear deformation theory. Compo. Struct. 94(6), (1). 36. Mori, T., Tanaka, K.: Average tre in matrix and average elatic energy of material with mifitting incluion. Acta Metall. 1(5), (1973). 37. Vel, S.S., Batra, R.C.: Three-dimenional exact olution for the viration of functionally graded rectangular plate. J. Sound Vi. 7(3-5), (4).

21 Figure Caption Fig. 1. Variation of exponential function f ( ) through the thickne of a FG plate for variou value of parameter p Fig.. Variation of dimenionle diplacement and tree through the thickne of plate ( a / h = 4, p =.5) Fig. 3. Variation of volume fraction V c through the thickne of the plate for variou value of the power law index p 1

22 Tale Caption Tale 1. Dimenionle deflection w ( ) of plate ( a / h = ) Tale. Dimenionle deflection w ( ) of plate ( a / h = 4) Tale 3. Dimenionle tre y ( h / ) of plate ( a / h = ) Tale 4. Dimenionle tre y ( h / ) of plate ( a / h = 4 ) Tale 5. Dimenionle fundamental frequency w of quare plate

23 p=1.5 /h Fig. 1. Variation of exponential function f ( ) through the thickne of a FG plate for variou value of parameter p f ( ) 3

24 /a= 1 3D [8] Preent 3D [8] Preent /h /h /a=1 u w 3D [8] Preent /a=1 /a= 1 3D [8] Preent /h /h x xy 3D [8] Preent 3D [8] Preent /h /a=1 /h /a= 1 x Fig.. Variation of dimenionle diplacement and tree through the thickne of plate ( a / h = 4, p =.5) y 4

25 p=5 p=1 p=5 Thickne coordinate, /h p= p=1 p=.5 p=. p=.1 p=. Fig. 3. Variation of volume fraction value of the power law index p Volume fraction, V c V c through the thickne of the plate for variou 5

26 Tale 1. Dimenionle deflection w ( ) of plate ( a / h = ) /a Method p D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35]

27 Tale. Dimenionle deflection w ( ) of plate ( a / h = 4 ) /a Method p D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35]

28 Tale 3. Dimenionle tre y ( h / ) of plate ( a / h = ) /a Method p D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35]

29 Tale 4. Dimenionle tre y ( h / ) of plate ( a / h = 4 ) /a Method p D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35] D [33] quai-3d [33] quai-3d [34] quai-3d [8] Preent HSDT [35]

30 Tale 5. Dimenionle fundamental frequency w of quare plate Method p= p=1 a/h=5 a / h = 1 a/h=1 a/h=5 a/h=1 a/h= p= p=3 p=5 3D [37] Quai-3D [6] Quai-3D [7] Quai-3D [4] TSDT [1] Preent