Free vibration of fg sandwich plates partially supported by elastic foundation using a quasi-3d finite element formulation

Nội dung text: Free vibration of fg sandwich plates partially supported by elastic foundation using a quasi-3d finite element formulation

1. Vietnam Journal of Mechanics, VAST, Vol.42, No. 1 (2020), pp. 63 – 86 DOI: FREE VIBRATION OF FG SANDWICH PLATES PARTIALLY SUPPORTED BY ELASTIC FOUNDATION USING A QUASI-3D FINITE ELEMENT FORMULATION Le Cong Ich1,∗, Pham Vu Nam2,3, Nguyen Dinh Kien3,4 1Le Quy Don Technical University, Hanoi, Vietnam 2Thuyloi University, Hanoi, Vietnam 3Graduate University of Science and Technology, VAST, Hanoi, Vietnam 4Institute of Mechanics, VAST, Hanoi, Vietnam ∗E-mail: ichlecong@gmail.com Received: 19 December 2019 / Published online: 25 March 2020 Abstract. Free vibration of functionally graded (FG) sandwich plates partially supported by a Pasternak elastic foundation is studied. The plates consist of three layers, namely a pure ceramic hardcore and two functionally graded skin layers. The effective material properties of the skin layers are considered to vary in the plate thickness by a power grada- tion law, and they are estimated by Mori–Tanaka scheme. The quasi-3D shear deformation theory, which takes the thickness stretching effect into account, is adopted to formulate a finite element formulation for computing vibration characteristics. The accuracy of the de- rived formulation is confirmed through a comparison study. The numerical result reveals that the foundation supporting area plays an important role on the vibration behavior of the plates, and the effect of the layer thickness ratio on the frequencies is governed by the supporting area. A parametric study is carried out to highlight the effects of material distribution, layer thickness ratio, foundation stiffness and area of the foundation support on the frequencies and mode shapes of the plates. The influence of the side-to-thickness ratio on the frequencies of the plates is also examined and discussed. Keywords: FG sandwich plate, Pasternak foundation, Mori–Tanaka scheme, quasi-3D the- ory, free vibration, finite element formulation. 1. INTRODUCTION Sandwich structures with high rigidity, low specific weight, excellent vibration char- acteristics and good fatigue properties have great potential for use in aerospace industry. These structures, usually consist of a core bonded to two skin layers, however encounter the delamination due to the sudden change in the material properties from one layer to another. Thanks to the advanced manufacturing methods [1], functionally graded mate- rials initiated by Japanese scientists in mid-1980 can now be incorporated into sandwich c 2020 Vietnam Academy of Science and Technology
2. 64 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien construction to improve performance of the structures. Functionally graded (FG) sand- wich structures can be designed to have a smooth variation of the properties, and this helps to avoid the delaminating problem. Many investigations on the mechanical behav- ior of FG and FG sandwich plates, the structures considered in this paper, are summa- rized in the review papers [2,3], the contributions that are most relevant to the present work are briefly discussed below. Praveen and Reddy [4] took the effect of temperature rise into consideration in their derivation of a first-order shear deformable four-node quadrilateral (Q4) element for non- linear transient analysis of FG plates. Zenkour [5,6] presented a sinusoidal shear defor- mation plate theory for bending, buckling and vibration analyses of FG sandwich plates. The effect of the material distribution, side-to-thickness ratio, core thickness on the fre- quencies are illustrated by the author through a simply supported plate. The theory was then employed by Zenkour and Sobhy [7] to study the thermal buckling of FG sandwich plates with temperature-dependent material properties. A n-order shear deformation theory was proposed by Xiang et al. [8] for free vibration analysis of FG sandwich plates. Zero transverse shear stresses at the top and bottom surfaces of plates are satisfied in the theory, and the Reddy’s third-order shear deformation theory can be obtained as a spe- cial case. The n-order shear deformation theory was then used in combination with the meshless global collocation method by Xiang et al. [9] to compute the frequencies of FG sandwich plates. Neves et al. [10] derived a quasi-3D shear deformation theory for ana- lyzing isotropic and FG sandwich plates by taking the extensibility in the thickness direc- tion into account. The collocation with radial basis functions was adopted by the authors to obtain the static and free vibration characteristics of the plates. Various higher-order shear deformation theories for analysis of FG plates were proposed by Thai and his co- workers in [11–13]. In the theories, the transverse displacement is split into two parts, the bending and shear parts. In [14], Thai et al. proposed a new first-order shear deformation theory for analysis of sandwich plates with an isotropic homogeneous core and two FG face layers. The shear stresses in the theory are directly computed from transverse shear forces, and shear correction factors are not necessary to use. Iurlaro et al. [15] adopted the refined zigzag theory to formulate finite element formulations for bending and free vibra- tion analysis of FG sandwich plates. The numerical investigations by the authors showed that the zigzag theory is superior in predicting the mechanical behavior of the plates to the first-order and third-order shear deformation theories. Pandey and Pradyumna [16] employed the higher-order layerwise theory to derive an eight-node isoparametric el- ement for static and dynamic analyses of FG sandwich plates. The numerical results obtained in the work showed the efficiency and accuracy of the derived element in eval- uating the bending and dynamic chracteristices of the plates. Belabed et al. [17] pro- posed a three-unknown hyperbolic shear deformation theory for free vibration study of FG sandwich plates with a homogeneous or FG core. Recently, Daikh and Zenkour [18] considered the effect of porosities in bending behavior of FG sandwich plates. Power law and sigmoid functions are adopted by the authors to describe the variation of the material properties of the FG skin layers. The effect of elastic foundation support on mechanical behavior of FG and FG sand- wich plates has been reported by several authors. In this line of works, Luă et al. [19]
3. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 65 considered the interaction between plate surface and foundation as the traction bound- ary conditions of the plate in their free vibration analysis of an FG plate resting on a Pasternak foundation. By expanding the state variables in trigonometric dual series and with the aid of the state space method, the authors obtained an exact solution for a sim- ply supported plate. Also adopting the Pasternak foundation model, Benyoucef et al. [20] studied reponse of a simply supported FG plate on foundation to distributed loads. Equi- librium equations were derived using the hyperbolic shear deformation theory and the Navier solution was employed to obtain the displacements. Various shear deformation theories were employed by Sobhy [21] to study buckling and free vibration of FG sand- wich plates resting on a Pasternak foundation. The effects of Winkler and Pasternak foundation parameters on bending of FG plates were considered by Al Khateeb and Zenkour [22], taking the influence of temperature and moisture into account. The influ- ence of tangential edge constraints and foundation support on buckling and postbuck- ling behaviour of FG sandwich plates and FG sandwich spherical shells was respectively considered by Tung [23], Khoa and Tung [24] using the Galerkin method. Based on a hyperbolic shear and normal deformation plate theory, Akavci [25] carried out static bending, buckling and free vibration analyses of FG sandwich plates supported by a Pasternak foundation. In [26], the effect of neutral surface position was taken into ac- count in studying vibration of a rectangular FG plate resting on an elastic foundation. Bending and vibration analyses of FG plates on an elastic foundation were performed by Benahmed et al. [27] using a quasi-3D hyperbolic shear deformation theory. Free vibra- tion of FG plates on a Pasternak foundation was recently investigated through 2D and quasi-3D shear deformation theories [28]. It has been shown that the frequencies and mode shapes of structures partially sup- ported by an elastic foundation are much different from that of the ones fully supported by the foundation [29, 30]. The vibration modes of plates, as shown by Motaghian et al. in [31], are governed by the area and position of the foundation support as well. To the authors’ best knowledge, the free vibration of FG sandwich plates partially supported by an elastic foundation has not been reported in the literature and it is considered in the present paper. The plates considered herein are composed of three layers, a ceramic core and two FG skin layers. The material properties of the skin layers are assumed to vary in the thickness direction by a simple power gradation law, and they are estimated by Morri–Tanaka scheme. Pasternak foundation model is adopted herein for describing the foundation. Based on a quasi-3D shear deformation theory, a finite element formulation is derived and employed to compute frequencies and mode shapes of the plates. The ef- fects of the material distribution, layer thickness ratio and foundation parameters on the vibration characteristics are investigated in detail. The influence of the side-to-thickness ratio on the frequencies is also examined and discussed. 2. MATHEMATICAL MODEL Fig.1 shows a rectangular FG sandwich plate with length a, width b and thickness h, partially supported by an elastic foundation. The Cartersian coordinate system (x, y, z) in the figure is chosen such that the (x, y) plane is coincident with the mid-plane, and z-axis directs upward.
4. parameters on the vibration characteristics are investigated in detail. The influence of the side-to- thickness ratio on the frequencies is also examined and discussed. parameters on the vibration characteristics2. MATHEMATICAL are investigated in detail MODEL. The influence of the side-to- thickness ratio on the frequencies is also examined and discussed. Fig. 1 shows a rectangular FG sandwich plate with length a, width b and thickness h, partially supported by an elastic foundation. The Cartersian coordinate system (x,y,z) in the figure is chosen such 2. MATHEMATICAL MODEL that the (x,y) plane is coincident with the mid-plane, and z-axis directs upward. Fig. 1 shows a rectangular FG sandwich plate with length a, width b and thickness h, partially 66 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien supported by an elastic foundation. yThe Cartersian coordinate system (x,y,z) in the figure is chosen such that the (x,y) plane is coincident with the mid-plane, and z-axis directs upward. y a y a h b y (,)ab z b h (,)ab z (,)abff z3 (,)ab z ff z2 3 x z2 x z1 z1 z0 z Shear layer (k ) 0 Shear layer (k1 ) 1 x x WinklerWinkler layer layer (k0 )(k0 ) (0,0) (0,0) (a) (b) (a)(a) (b) (b) Figure. 1. FG sandwich plate partially supported by a Pasternak elastic foundation Fig.Fig 1ure. FG. 1 sandwich. FG sandwich plate plate partially partially supported supporte byd a by Pasternak a Pasternak elastic elastic foundation foundation The plate consists of three layers, a homogeneous ceramic core and two FG metal-ceramic skin layers.The Denotingplate consists z0, z1, z2of and three z3 are, layer respectively,s, a homogeneous the vertical ordinates ceramic of corethe bottom and twosurface, FG the metal two -ceramic skin layer interfaces and the top surface, in which z = - h/2 and z = h/2. The foundation considered herein is layers. DenotingThe plate z0, z consists1, z2 and ofz3 threeare, respectively, layers,0 a homogeneous the3 vertical ordinates ceramic of core the andbottom two surface, FG metal- the two a Pasternak model, which consists of elastic springs with stiffnes k0 and a shear layer with stiffness k1. layer interfaces and the top surface, in which z0 = - h/2 and z3 = h/2. The foundation considered herein is Theceramic foundation skin area layers. is assumed Denoting to be rectangularz0, z1 ,withz2 andlengthz 3af are,and width respectively, bf, supported the the plate vertical at its ordinatesleft of a Pasternak model, which consists of elastic springs with stiffnes k0 and a shear layer with stiffness k1. cornerthe as bottom shown in surface, Fig. 1(b) the. The two volume layer fraction interfaces of the constituents and the of top the surface,skin layers in is whichsupposedz to0 = −h/2 Thevary foundation in the thickness area isdirection assumed according to be torectangular with length af and width bf, supported the plate at its left and z3 = h/2. The foundation considered herein is a Pasternak model, which consists of corner as shown in Fig. 1(b). Then volume fraction of the constituents of the skin layers is supposed to elastic springsỡ withổửzz stiffness- k0 and a shear layer with stiffness k1. The foundation area vary in the thicknessùVz(1) ()direction= according0 ,z ẻto zz, c ỗữ [ 01] a b is assumedù to be rectangularốứzz10- with length f and width f , supported the plate at its left ù n corner as shownỡ (2) inổử Fig.zz- 1(b). The volume fraction of the constituents of the skin layers is ớVzc(1) ()= 1 0 ,zẻ[zz12, ] (1) ùVzc ()=ỗữ,zẻ[zz01, ] supposed toù vary in the thicknessn direction according to ù ốứzz10- ù (3) ổửzz- 3 ()kk () ù Vcm()zz=ỗ ữ ,aẻ[zz2 ,13 ] nd V=- Vc ù (2) zz-  n ớợVzc ()=ốứ 12 3 (1) ,zẻz[−zz12,z0 ] (1) Vc (z) = , z ∈ [z0, z1] where k=1, 2, 3;ù V and V are, respectively,n thez volume− z fraction of the metal and ceramic; n is the ù m ổửc zz- 1 0 (3) 3(2) ()kk () power-law materialVcm index,()zz= definingỗ theữ variation,a ofẻ constituents[zz2 ,13 ] throughnd Vthe plate= -thickness. Vc ù Vc (z) = 1 , z ∈ [z1, z2] ợ ốứzz2 - 3 Mori-Tanaka scheme is employed herewith to  estimaten the effective material properties. (1) − ()k ()k According to the Mori-Tanaka scheme,(3) the effectivez localz3 bulk modulus K and shear modulus G of where k=1, 2, 3; Vm and Vc are,Vc respectively,(z) = the volume , fractionz ∈ f[z 2 of, z 3 the] metal and f ceramic; n is the th z − z powerthe -klaw layer material of the sandwich index, definingplate can bethe given variation by [322 ] of constituents3 through the plate thickness. KK()kk- () (k) ()k (k) fm V = 1 −VcV , Mori-Tanaka scheme= is employedm herewithc to estimate; the effective material(2) properties. KK()kk- () ()kkk () () () k () k cm11+( VKKccm)( ) /( K m+ 4 G m /3) ()k ()k Accordingwhere to()kkk the= () Mori1, 2,- 3;TanakaV and scheme,V are, the respectively, effective local the bulk volume modulus fraction K f and of theshear metal modulus and G ce-f of GG- m c V ()k th fmn = c the k ramic; layer ()ofkk theis () the sandwich power-law()kkk plate () can material () be given () kk index, () by [ ()32defining k] () k the() k variation () k of constituents(3) through GGcm- 11+ VGG/ GGK+ 9 + 8 G /6ộự K+ 2 G ()kk( ()ccm)( ) { mmm( m) ( m m) } the plate thickness.KK- Mori–Tanaka schemeV () isk employedởỷ herewith to estimate the effec- fm= c ; wheretive material() properties.kk () According()kkk to () the Mori–Tanaka () () k scheme, () k the effective local bulk(2) (KK) - 11+ VKK( ) / K+ 4 G /3 modulus K k cmand shear modulus( ccmG)( k of the kth) layer( m of the m sandwich) plate can be given ()kk ()f f 3 GG- V ()k by [32fm] = c ()kk () ()kkk () () () kk () () k () k() k () k (3) GGcm- 11(+k) VGG(k) / GGK+ 9 + 8 G /6ộự K+ 2 G ( ccm)( ) { mmm( (k) m) ( m m) } K f − Km V ởỷ = c , (2) where (k) (k)  (k)  (k) (k)  (k) (k)  Kc − Km 1 + 1 − Vc Kc − Km / Km + 4Gm /3 3
5. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 67 (k) (k) (k) Gf − Gm V = c , (k) (k)  (k)  (k) (k) n (k) (k)  (k) (k) h  (k) (k)io (3) Gc − Gm 1 + 1 − Vc Gc − Gm / Gm + Gm 9Km + 8Gm / 6 Km + 2Gm where (k) (k) (k) (k) ( ) E ( ) E ( ) E ( ) E K k = c , G k = c , K k = m , G k = m m, c  (k) c  (k) m  (k) m  (k) 3 1 − 2àc 2 1 + àc 3 1 − 2àm 2 1 + àm (4) are the local bulk and the shear moduli of the ceramic and metal at the kth layer, respec- tively. (k) Noting that the effective mass density ρ f is defined by Voigt model as (k) (k) (k) (k) (k) ρ f = (ρc − ρm )Vc + ρm . (5) (k) (k) The effective Young’s modulus Ef and Poisson’s ratio υ f are computed via effec- tive bulk modulus and shear modulus as (k) (k) (k) (k) ( ) 9K f Gf ( ) 3K f − 2Gf E k = , υ k = . (6) f (k) (k) f (k) (k) 3K f + Gf 6K f + 2Gf Based on the quasi-3D shear deformation theory [12,13], the displacements in the x-, y- and z-directions, u(x, y, z, t), v(x, y, z, t) and w(x, y, z, t), are, respectively, given by u (x, y, z, t) = u0(x, y, t) − zwb,x(x, y, t) − f (z) ws,x(x, y, t), v (x, y, z, t) = v0(x, y, t) − zwb,y(x, y, t) − f (z) ws,y(x, y, t), (7) w (x, y, z, t) = wb(x, y, t) + ws(x, y, t) + g (z) wz(x, y, t), where u0(x, y, t) and v0(x, y, t) are, respectively, the displacements in x- and y-directions of a point on the mid-plane; wb(x, y, t), ws(x, y, t) and wz(x, y, t) are, respectively, bending and shear components of the transverse displacement, and 4 4 f (z) = z3, g(z) = 1 − f = 1 − z2. (8) 3h2 ,z h2 In the above equation and hereafter, a subscript comma is used to denote the deriv- ative with respect to the followed variable, e.g. f,z = ∂ f /∂z. The strains resulted from Eq. (7) are of the forms    0   b   s  εx εx  κx  κx    0   b   s  εy = εy + z κy + f (z) κy ,    0   b   s  γxy γxy  κxy  κxy (9)     γxz κxz = g(z) , εz = g,z (z) wz, γyz κyz
6. 68 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien where 0 0 0 b b εx = u0,x , εy = v0,y , γxy = u0,y + v0,x , κx = −wb,xx , κy = −wb,yy , b s s s κxy = −2wb,xy , κx = −ws,xx , κy = −ws,yy , κxy = −2ws,xy , (10) κxz = (ws,x + wz,x) ,, κyz = (ws,y + wz,y). The constitutive equations based on linear behaviour of the plate material are of the forms  (k) (k) (k)    Q Q Q 0 0 0   σ 11 12 12 ε  x   ( ) ( ) ( )   x     Q k Q k Q k     σy   12 11 12 0 0 0   εy           (k) (k) (k)    σz  Q Q Q 0 0 0  εz =  12 12 11  , (11)  τxy   (k)   γxy     0 0 0 Q 0 0     τ   44   γ   xz   (k)   yz     0 0 0 0 Q44 0    τyz (k) γxz 0 0 0 0 0 Q44 in which 2 (k)  (k) − ( (k)) υ 1 + υ (k) (k) (k) 1 υ f (k) (k) f f (k) (k) (k) Ef Q = E , Q = E , Q = G , G = . 11 f 2 3 12 f 2 3 44 f f (k) (k) (k) (k) (k) 2(1 + υ ) 1 − 3(υ f ) − 2(υ f ) 1 − 3(υ f ) − 2(υ f ) f (12) The strain energy stemming from the plate deformation is given by 1 Z UP = σ ε + σ ε + σ ε + τ γ + τ γ + τ γ
   dV, (13) 2 x x y y z z xy xy yz yz xz xz V with V is the volume of the plate. Substituting Eqs. (9)–(11) into Eq. (13), one gets a b 1 Z Z n      64  UP = A u2 + v2 − 2A u w + v w + A w2 + w2 + w2 2 11 0,x 0,y 12 0,x b,xx 0,y b,yy 22 b,xx b,yy h4 z 0 0 8
   8   16  2 2  − A u ws,xx + v ws,yy + A w ws,xx + w ws,yy + A w + w 3h2 23 0,x 0,y 3h2 44 b,xx b,yy 9h4 66 s,xx s,yy   8
   8
   + 2B u v − 2B u w + v w + u + v wz − B u ws yy + v ws xx 11 0,x 0,y 12 0,x b,yy 0,y b,xx h2 0,x 0,y 3h2 23 0,x , 0,y ,   8   8
   + 2B w w + w + w wz − B u ws yy + y ws xx 22 b,xx b,yy h2 b,xx b,yy 3h2 23 0,x , 0,y ,   8   64
   32 + B w ws,yy + w ws,xx + ws,xx + ws,yy wz + B ws,xxws,yy 44 3h2 b,xx b,yy 3h4 9h4 66 h
   2 2 2 2 2
   i
   + C11 u0,y + v0,x + wz,x + wz,y + ws,x + ws,y + 2 ws,xwz,x + ws,ywz,y − 4C12 u0,y + v0,x wb,xy  2   4 2   + 4C w2 − w2 + w2 − w w + w w
   − w2 + w2 22 b,xy h2 s,x s,y h2 s,x z,x s,y z,y h2 z,x z,y   16 2 2 2 2
   2 2 + C w + w + w + w + 2 ws xwz x + ws ywz y + h w ws xy h4 44 s,x s,y z,x z,y , , , , 3 b,xy ,  16
   64 2 − C u + v ws,xy + C w dxdy. 3h2 23 0,y 0,x 9h4 66 s,xy (14)
7. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 69 In the above equation, A11, A12, , C44, C66 are the plate rigidities, defined as z 3 Z k   = (k) 2 3 4 6 (A11, A12, A22, A23, A44, A66) ∑ Q11 1, z, z , z , z , z dz, k=1 zk−1 z 3 Z k   = (k) 2 3 4 6 (B11, B12, B22, B23, B44, B66) ∑ Q12 1, z, z , z , z , z dz, (15) k=1 zk−1 z 3 Z k   = (k) 2 3 4 6 (C11, C12, C22, C23, C44, C66) ∑ Q44 1, z, z , z , z , z dz. k=1 zk−1 The strain energy resulted from the foundation deformation is of the form ZZ h  i F = 2 + 2 + 2 U k0w0 k1 w0,x0,x w0,y dS SF a f b f Z Z n 2 h 2 2io = k0(wb + ws + wz) + k1 (wb,x + ws,x + wz,x) + (wb,y + ws,y + wz,y) dxdy, 0 0 (16) where SF is the area of the foundation support, and w0 = w(z = 0). The total energy U of the plate with the foundation support is U = UP + UF. (17) The kinetic energy of the plate resulted from Eq. (7) is of the form a b 1 Z Z n h i T = I u˙2 + v˙2 + (w˙ + w˙ )2 + 2I u˙ w˙ + v˙ w˙
   2 11 0 0 b s 12 0 b,x 0 b,y 0 0  8  8 (18) + I w˙ 2 + w˙ 2 − (w˙ + w˙ + w˙ ) w˙ − I u˙ w˙ + v˙ w˙
   12 b,x b,y h2 b s z z 3h2 23 0 s,x 0 s,y 8  2  16   − I w˙ w˙ + w˙ w˙ + w˙ 2 + w˙ 2 + w˙ 2 dxdy, 3h2 44 b,x s,x b,y s,y h2 z 9h4 s,x s,y where the mass moments I11, I12, , I66 are defined as z 3 Z k h i ( ) = (k) 2 3 4 6 I11, I12, I22, I23, I44, I66 ∑ ρ f 1, z, z , z , z , z dz, (19) k=1 zk−1 (k) where the effective mass density ρ f is defined by Eq. (5). Equations of motion for the plate can be obtained by applying Hamilton’s princi- ple to Eqs. (17) and (18). However, a closed-form solution for such equations is hardly obtained for the plate partially supported by the elastic foundation. A finite element for- mulation is derived in the next section for obtaining frequencies and vibration modes of the plate.
8. 70 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien 3. FINITE ELEMENT FORMULATION A Q4 plate element with size of (xe, ye) is derived in this section. In addition to the values of the displacements at the nodes, their derivatives are also taken as degrees of freedom, and the vector of nodal displacements is given by =  T d du0 dv0 dwb dws dwz , (20) (44ì1) where and hereafter, a superscript ‘T’ denotes the transpose of a vector or a matrix; du0 , dv0 , dwb , dws and dwz are defined as  1 2 3 4 T  1 2 3 4 T  1 2 3 4 T du0 = u0 u0 u0 u0 , dv0 = v0 v0 v0 v0 , dwz = wz wz wz wz , (21) and n o n o = 1 2 3 4 T = 1 2 3 4 T dwb dwb dwb dwb dwb , dws dws dws dws dws , (22) with n oT n oT j = j j j j j = j j j j = dwb wb wb,x wb,y wb,xy , dws ws ws,x ws,y ws,xy , j 1, . . . , 4. (23) The stiffness and mass matrices for the element are better to derived in term of the natural coordinates ξ and η: ξ = 2(x − xC)/xe, η = 2(y − yC)/ye, with (xC, yC) is the centroid coordinates of the element. For −xe/2 ≤ (x − xC) ≤ xe/2 ⇒ −1 ≤ ξ ≤ 1 and −ye/2 ≤ (y − yC) ≤ ye/2 ⇒ −1 ≤ η ≤ 1 [33]. In this regard, the relation between the derivatives in the two coordinate systems are given by ∂(.) ∂(.) ∂x ∂(.) ∂y = + ∂ξ ∂x ∂ξ ∂y ∂ξ  (.)   (.)   (.)   (.)  or ,ξ = J ,x and ,x = J−1 ,ξ , ∂(.) ∂(.) ∂x ∂(.) ∂y (.),η (.),y (.),y (.),η = + ∂η ∂x ∂η ∂y ∂η (24) with the Jacobian matrix J is of the form  x y  J = ,ξ ,ξ . (25) x,η y,η The displacements u0, v0 and wz are interpolated from their nodal values as 4 4 4 = = i = = i = = i u0 Ndu0 ∑ Niu0 , v0 Ndv0 ∑ Niv0 , wz Ndwz ∑ Niwz , (26) i=1 i=1 i=1 where Ni are the Lagrangian functions with the following form 1 N = (1 + ξ ξ)(1 + η η) (i = 1, . . . , 4) and N =  N N N N  , (27) i 4 i i 1 2 3 4
9. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 71 As seen from Eqs. (9) and (10), the transverse bending and shear displacements should be twice differentiable, and Hermite polynomials are employed herein to inter- polate these displacements as 4 4 h i n oT w = H di = H1 H2 H3 H4 wi wi wi wi , b ∑ i wb ∑ i i i i b b,x b,y b,xy i= i= 1 1 (28) 4 4 h i n oT w = H di = H1 H2 H3 H4 wi wi wi wi s ∑ i ws ∑ i i i i s s,x s,y s,xy , i=1 i=1 j where the interpolation functions Hi have the following forms [34] 1 H1 = (ξ + ξ )2 (ξξ − 2)(η + η )2 (ηη − 2) , i 16 i i i i 1 H2 = ξ (ξ + ξ )2 (1 − ξξ )(η + η )2 (ηη − 2) , i 16 i i i i i 1 H3 = − (ξ + ξ )2 (ξξ − 2) η (η + η )2 (ηη − 1) , (29) i 16 i i i i i 1 H4 = ξ (ξ + ξ )2 (ξξ − 1) η (η + η )2 (ηη − 1) , i 16 1 i i i i i  1 2 3 4    Hi = Hi Hi Hi Hi , i = (1, . . . , 4), H = H1 H2 H3 H4 . 1ì16 Using the above interpolation scheme, one can write the strain energy Ue of the ele- ment in terms of the nodal displacement vector (d) as NEP NEF 1 T P 1 T F Ue = ∑ di ki di + ∑ di ki di, (30) 2 i=1 2 i=1 where ‘NEP’ and ‘NEF’ are, respectively, the total numbers of elements used to discrete the plate and the foundation; kP and kF are, respectively, the element stiffness matrices resulted from the plate and the foundation deformation. The stiffness matrix kP can be written in sub-matrices as  P P P P P  kuu kuv kuwb kuws kuwz T   P  P P P P   kuv kvv kvw kvws kvwz   b    T  T  P  kP kP kP kP kP  k =  uwb vwb wbwb wbws wbwz  , (31)   T  T  T   kP kP kP kP kP   uws vws wbws wsws wswz   T T T T   P   P   P   P  P kuwz kvwz kwbwz kwswz kwzwz where the sub-matrices have the following forms Z1 Z1 P  T T  kuu = N,xA11N,x + N,yC11N,y |J| dξdη, (32a) 4ì4 −1 −1
10. 72 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien 1 1 Z Z P  T T  kvv = N,yA11N,y + N,xC11N,x |J| dξdη, (32b) 4ì4 −1 −1 1 1 Z Z h i P = T + T + + T + T | | kwb wb H,xxA22H,xx H,yyA22H,yy 2H,xxB22H,yy 4H,xyC22H,xy J dξdη, (32c) ì 16 16 −1 −1 1 1 1 Z Z   kP = [16 HT A H + HT A H + 32HT B H + 64HT C H wsws 4 ,xx 66 ,xx ,yy 66 j,yy ,xx 66 ,yy ,xy 66 ,xy ì 9h 16 16 −1 −1 (32d)  T T   4 2   T T  + 144C44 H,xH,x + H,yH,y + 9h C11 − 72h C22 H,xH,x + H,yH,y ] |J| dξdη, 1 1 1 Z Z h    i kP = 16C − 8h2C + h4C NT A N + NT A N + 64NTA N |J| dξdη, wzwz 4 44 22 11 ,x 11 ,x ,y 11 ,y 22 (32e) ì h 4 4 −1 −1 1 1 Z Z P  T T  kuv = N,xB11N,y + N,yC11N,y |J| dξdη, (32f) 4ì4 −1 −1 1 1 Z Z   P = − T + T + T | | kuwb N,xA12H,xx N,xB12H,yy 2N,yC12H,xy J dξdη, (32g) ì 4 16 −1 −1 1 1 4 Z Z   kP = − NT A H + NT B H + 2NT C H |J| dξdη, uws 2 ,x 23 ,xx ,x 23 ,yy ,y 23 ,xy (32h) ì 3h 4 16 −1 −1 1 1 8 Z Z   kP = − NTB N |J| dξdη, uwz 2 12 ,x (32i) ì h 4 4 −1 −1 1 1 Z Z   P = − T + T + T | | kvwb N,yA12H,yy N,yB12H,xx 2N,xC12H,xy J dξdη, (32j) ì 4 16 −1 −1 1 1 4 Z Z   kP = − NT A H + NT B H + 2NT C H |J| dξdη, vws 2 ,y 23 ,yy ,y 23 ,xx ,x 23 ,xy (32k) ì 3h 4 16 −1 −1 1 1 8 Z Z   kP = − NTB N |J| dξdη, vwz 2 12 ,y (32l) ì h 4 4 −1 −1 1 1 4 Z Z   kP = HT A H + HT A H + 2HT B H + 4HT C H |J| dξdη, wb ws 2 ,xx 44 ,xx ,yy 44 ,yy ,xx 44 ,yy ,xy 44 ,yy (32m) ì 3h 16 16 −1 −1 1 1 8 Z Z   kP = HT B N + HT B N |J| dξdη, wb wz 2 ,xx 22 ,yy 22 (32n) ì h 16 4 −1 −1 1 1 1 Z Z     kP = [32 HT B N + HT B N − 24h2 HT C N + HT C N wbwz 4 ,xx 44 ,yy 44 ,x 22 ,x ,y 22 ,y 16ì4 3h −1 −1 (32o) 1 1 Z Z 1     + kP = [+ 3h4C + 48B HT N + HT N ] |J| dξdη, wb wz 4 11 44 ,x ,x ,y ,y ì 3h −1 −1 16 4
11. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 73 with |J| = det(J). The element stiffness matrix stemming from the foundation deformation is of the form  0 0 0 0 0   0 0 0 0 0     0 0 kF kF kF  F  wbwb wbws wbwz  k =   T  , (33)  0 0 kF kF kF   wbws wsws wswz  T T   F   F  F  0 0 kwbwz kwswz kwzwz where 1 1 Z Z   F = F = F = Tk + T k + T k | | kwbwb kwsws kwbws H 0H H,x 1H,x H,y 1H,y J dξdη, (34a) ì 16ì16 ì 16 16 16 16 −1 −1 1 1 Z Z   F = T + T + T | | kwzwz N k0N N,xk1N,x N,yk1N,y J dξdη, (34b) 4ì4 −1 −1 1 1 Z Z   F = F = Tk + T k + T k | | kwbwz kwswz H 0N H,x 1N,x H,y 1N,y J dξdη. (34c) ì 16ì4 16 4 −1 −1 Similarly, the kinetic energy can be written in the following form NEp 1 ˙ T ˙ T = ∑ di midi, (35) 2 i where d˙ = d,t, and the element mass matrix m is defined as   muu 0 muwb muws 0 0 m m m 0  vv vwb vws   T T  m =  muwb mvwb mwbwb mwbws mwbwz  , (36)  T T T   muws mvws mwbws mwsws mwswz  T T 0 0 mwbwz mwswz mwzwz where the sub-matrices have the following forms Z1 Z1 T muu = mvv = N I11N |J| dξdη, (37a) 4ì4 4ì4 −1 −1 1 1 Z Z   = T + T + T | | mwbwb H I11H H,xI22H,x H,yI22H,y J dξdη, (37b) ì 16 16 −1 −1
12. 74 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien 1 1 Z Z  16   m = HT H + HT H + HT H |J| ξ η wsws I11 4 ,xI66 ,x ,yI66 ,y d d , (37c) 16ì16 9h −1 −1 1 1 Z Z  8 16  m = NT N − NT N + NT N |J| ξ η wzwz I11 2 I22 4 I44 d d , (37d) 16ì16 h h −1 −1 1 1 Z Z   = − T | | muwb N I12H,x J dξdη, (37e) ì 4 16 −1 −1 1 1 4 Z Z   m = − NT H |J| ξ η uws 2 I23 ,x d d , (37f) 4ì16 3h −1 −1 1 1 Z Z   = − T | | mvwb N I12H,y J dξdη, (37g) ì 4 16 −1 −1 1 1 4 Z Z   m = − NT H |J| ξ η vws 2 I23 ,y d d , (37h) 4ì16 3h −1 −1 1 1 Z Z  4   m = HTI H + HT I H + HT I H |J| dξdη. (37i) wbws 11 2 ,x 44 ,x ,y 44 ,y ì 3h 16 16 −1 −1 Since the highest order of the polynomials under the integrals in Eqs. (33) and (36) is six, and thus 4-Gauss point along the ξ and η directions is enough to evaluate the integrals. Having the derived element stiffness and mass matrices, the equation of motion for free vibration analysis of the plate can be written in the following form MDă + KD = 0, (38) where D and Dă are, respectively, the structural vectors of nodal displacements and ac- celerations; M and K are the structural mass and stiffness matrices of the plate-elastic foundation system, obtained by assembling the above derived element mass and stiff- ness matrices, respectively. For free vibration problems, Eq. (38) can be expressed as the following eigenvalue problem, which can be solved in the standard way to obtain natural frequencies and mode shapes of the plate [K] − ω2 [M]
    {X} = 0, (39) where ω is the eigenfrequency, {X} is the generalized eigenvector.
13. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 75 4. NUMERICAL RESULTS AND DISCUSSION The vibration characteristics of the FGSW plate partially supported by the elastic foundation are reported in this section. Otherwise stated, the plate formed from Alu- mina (Al2O3) and Aluminum (Al) with the following properties [25] is employed in the analysis 3 - Alumina Al2O3 (ceramic): Ec = 380 GPa; νc = 0.3; ρc = 3800 kg/m . 3 - Aluminum (Al) (metal): Em = 70 GPa; νm = 0.3; ρm = 2707 kg/m . For convenience of discussion, the following non-dimensional frequency parameter and foundation stiffness parameters are used [25] 2
    p 4 2 ω¯ = ωa h ρ0/E0, Kw = k0a /DC, Ks = k1a /DC, (40) 3 2 where ω is the fundamental frequency, and DC = Ech /[12(1 − v )], E0 = 1 GPa, ρ0 = 1 kg/m3. Three number in brackets are used herein to denote the layer thickness ratio, e.g. (1-2-1) means that the thickness ratio of the bottom layer, the core layer and the top layer is 1:2:1. Three types of boundary conditions, namely simply supported at all edges (SSSS), simply supported at two opposite edges and clamped at the others (SCSC) and clamped at all edges (CCCC), are considered herein. The constraints for these boundaries are as follows: - For simply supported edge: + u0 = wb = ws = wz = wb,y = ws,y = wb,xy = ws,xy = 0 at x = 0, a. + v0 = wb = ws = wz = wb,x = ws,x = wb,xy = ws,xy = 0 at y = 0, b. - Clamped egde: u0 = v0 = wb = ws = wz = wb,x = wb,y = ws,x = ws,y = wb,xy = ws,xy = 0. 4.1. Formulation verification Since the data for the FGSW plate partially supported by the elastic foundation are not available in the literature, the verification is carried out herewith by comparing the frequency parameters obtained in the present work with the published data as shown in Tab.1 for a simply supported FGSW plate fully supported by the elastic foundation. For both the side-to-thickness ratios, Tab.1 shows a good agreement between the result of the present work with that of Ref. [25], regardless of the material grading indexes, the foundation stiffness parameters and the layer thickness ratio. Noting that the plate used to obtain the result in Tab.1 is formed from Aluminum and Zirconia as employed in [25]. In addition, the convergence of the present formulation in evaluating the frequencies in Tab.1 has been achieved by using 20 elements, and this number of elements is used in all computations reported below. 4.2. Simply supported plate The frequency parameters of the SSSS square FGSW plate partially resting on the elastic foundation are respectively listed in Tabs.2,3 and4 for different values of the foun- dation stiffness parameters, the layer thickness ratio, and different foundation support- ing areas, namely (a f , b f ) = (a/4, b/4), (a f , b f ) = (a/2, b/2) and (a f , b f ) = (3a/4, 3b/4). As in case of the plate without or fully foundation support, the frequency parameter in the table shows a decrease by the increase of the material grading indexes, regardless
14. 76 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien Table 1. Comparison of frequency parameter of simply SSSS square plate fully supported by elastic foundation a/h N Kw Ks Theory (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-0-1) Akavci [25] 1.1912 1.1912 1.1912 1.1912 1.1912 0 0 Present work 1.1985 1.1985 1.1985 1.1985 1.1985 Akavci [25] 1.5135 1.5135 1.5135 1.5135 1.5135 0 10 10 Present work 1.5400 1.5400 1.5400 1.5400 1.5400 Akavci [25] 3.0908 3.0908 3.0908 3.0908 3.0908 100 100 Present work 3.0980 3.0980 3.0980 3.0980 3.0980 Akavci [25] 0.9318 0.9541 0.9755 0.9927 0.9088 0 0 Present work 0.9211 0.9433 0.9682 0.9832 0.8993 Akavci [25] 1.3341 1.3469 1.3611 1.3713 1.3231 2 10 10 5 Present work 1.3531 1.3650 1.3809 1.3892 1.3437 Akavci [25] 2.6823 2.7579 2.7937 2.8476 2.5621 100 100 Present work 2.6425 2.7274 2.7667 2.8278 2.5068 Akavci [25] 0.8791 0.8969 0.9215 0.9356 0.8633 0 0 Present work 0.8818 0.8992 0.9270 0.9379 0.8659 Akavci [25] 1.3045 1.3119 1.3274 1.3339 1.3022 10 10 10 Present work 1.3343 1.3410 1.3576 1.3617 1.3320 Akavci [25] 2.5044 2.6178 2.6707 2.7495 2.3176 100 100 Present work 2.4944 2.6115 2.6647 2.7471 2.3013 Akavci [25] 1.3404 1.3404 1.3404 1.3404 1.3404 0 0 Present work 1.3512 1.3512 1.3512 1.3512 1.3512 Akavci [25] 1.6590 1.6590 1.6590 1.6590 1.6590 0 10 10 Present work 1.6678 1.6678 1.6678 1.6678 1.6678 Akavci [25] 3.3694 3.3694 3.3694 3.3694 3.3694 100 100 Present work 3.3740 3.3740 3.3740 3.3740 3.3740 Akavci [25] 1.0182 1.0428 1.0694 1.0885 0.9971 0 0 Present work 1.0076 1.0311 1.0620 1.0778 0.9898 Akavci [25] 1.43 1.4444 1.4623 1.4740 1.4200 2 10 10 100 Present work 1.4225 1.4361 1.4569 1.4662 1.4149 Akavci [25] 3.3344 3.3283 3.3300 3.3261 3.3491 100 100 Present work 3.3315 3.3250 3.3279 3.3229 3.3472 Akavci [25] 0.9602 0.9758 1.0062 1.0191 0.9580 0 0 Present work 0.9657 0.9802 1.0143 1.0228 0.9651 Akavci [25] 1.3967 1.4029 1.4219 1.4278 1.4023 10 10 10 Present work 1.4005 1.4060 1.4277 1.4305 1.4073 Akavci [25] 3.3480 3.3332 3.3327 3.3225 3.3772 100 100 Present work 3.3499 3.3348 3.3354 3.3240 3.3795
15. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 77 Table 2. Frequency parameter of SSSS square plates partially supported by elastic foundation with (a f , b f ) = (a/4, b/4) a/h N Kw Ks (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-0-1) (1-8-1) 0 0 1.1886 1.2389 1.2867 1.3208 1.1343 1.5215 0.5 10 10 1.2349 1.2833 1.3294 1.3625 1.1830 1.5577 100 100 1.3905 1.4408 1.4865 1.5211 1.3344 1.7146 0 0 0.9637 1.0241 1.1006 1.1400 0.9159 1.4431 5 2 10 10 1.0211 1.0782 1.1510 1.1888 0.9766 1.4816 100 100 1.1660 1.2294 1.3030 1.3461 1.1095 1.6404 0 0 0.8934 0.9396 1.0267 1.0562 0.8678 1.4033 10 10 10 0.9565 0.9993 1.0814 1.1094 0.9330 1.4432 100 100 1.0930 1.1444 1.2287 1.2643 1.0481 1.6031 0 0 1.2590 1.3139 1.3688 1.4061 1.2033 1.6405 0.5 10 10 1.3054 1.3581 1.4112 1.4473 1.2521 1.6757 100 100 1.4936 1.5470 1.5992 1.6361 1.4388 1.8601 0 0 1.0096 1.0731 1.1586 1.1999 0.9666 1.5471 10 2 10 10 1.0682 1.1280 1.2094 1.2489 1.0290 1.5849 100 100 1.2472 1.3115 1.3936 1.4375 1.2021 1.7724 0 0 0.9351 0.9808 1.0773 1.1061 0.9295 1.4998 10 10 10 1.0000 1.0418 1.1328 1.1600 0.9976 1.5391 100 100 1.1727 1.2198 1.3128 1.3464 1.1625 1.7284 0 0 1.2861 1.3428 1.4007 1.4393 1.2298 1.6882 0.5 10 10 1.3325 1.3870 1.4430 1.4804 1.2787 1.7230 100 100 1.5345 1.5890 1.6439 1.6816 1.4808 1.9186 0 0 1.0268 1.0915 1.1806 1.2225 0.9860 1.5882 100 2 10 10 1.0858 1.1466 1.2315 1.2716 1.0489 1.6257 100 100 1.2797 1.3439 1.4294 1.4734 1.2410 1.8251 0 0 0.9507 0.9961 1.0963 1.1248 0.9538 1.5377 10 10 10 1.0163 1.0576 1.1521 1.1789 1.0230 1.5767 100 100 1.2053 1.2498 1.3463 1.3786 1.2147 1.7781 of the layer thickness ratio and the foundation stiffness. The decrease of the frequency parameter can be explained by the lower content of ceramic for the plate associated with a higher index n, as can be seen from Eq. (1). The tables also show an important role of the layer thickness ratio and the area of the foundation support on the frequency pa- rameter. A larger core thickness the plate has a higher frequency is, irrespective of the foundation supporting area and the foundation stiffness parameters. The effect of the layer thickness ratio, however influenced by the foundation support also. For example, with a/h = 5, n = 2, (Kw, Ks) = (10, 10), the frequency parameter increases 31.08% when
16. 78 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien Table 3. Frequency parameter of SSSS square plates partially supported by elastic foundation with (a f , b f ) = (a/2, b/2) a/h N Kw Ks (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-0-1) (1-8-1) 0 0 1.1886 1.2389 1.2867 1.3208 1.1343 1.5215 0.5 10 10 1.3976 1.4397 1.4810 1.5099 1.3536 1.6870 100 100 2.1598 2.2051 2.2451 2.2754 2.1064 2.4391 0 0 0.9637 1.0241 1.1006 1.1400 0.9159 1.4431 5 2 10 10 1.2200 1.2659 1.3286 1.3593 1.1878 1.6186 100 100 1.9519 2.0172 2.0863 2.1278 1.8823 2.3813 0 0 0.8934 0.9396 1.0267 1.0562 0.8678 1.4033 10 10 10 1.1745 1.2050 1.2741 1.2943 1.1638 1.5850 100 100 1.8799 1.9384 2.0210 2.0585 1.7962 2.3541 0 0 1.2590 1.3139 1.3688 1.4061 1.2033 1.6405 0.5 10 10 1.4659 1.5121 1.5599 1.5917 1.4206 1.8004 100 100 2.3095 2.3529 2.3967 2.4257 2.2658 2.6079 0 0 1.0096 1.0731 1.1586 1.1999 0.9666 1.5471 10 2 10 10 1.2669 1.3151 1.3857 1.4179 1.2399 1.7179 100 100 2.1021 2.1585 2.2322 2.2674 2.0648 2.5415 0 0 0.9351 0.9808 1.0773 1.1061 0.9295 1.4998 10 10 10 1.2185 1.2478 1.3249 1.3442 1.2277 1.6772 100 100 2.0432 2.0832 2.1704 2.1959 2.0384 2.5101 0 0 1.2861 1.3428 1.4007 1.4393 1.2298 1.6882 0.5 10 10 1.4924 1.5402 1.5908 1.6238 1.4466 1.8462 100 100 2.3656 2.4083 2.4539 2.4824 2.3261 2.6736 0 0 1.0268 1.0915 1.1806 1.2225 0.9860 1.5882 100 2 10 10 1.2845 1.3336 1.4074 1.4402 1.2598 1.7574 100 100 2.1593 2.2114 2.2868 2.3192 2.1376 2.6028 0 0 0.9507 0.9961 1.0963 1.1248 0.9538 1.5377 10 10 10 1.2350 1.2638 1.3441 1.3629 1.2531 1.7136 100 100 2.1073 2.1383 2.2267 2.2470 2.1442 2.5694 the core thickness changes from (2-1-2) to (1-8-1) for the plate supported by the founda- tion with (a f , b f ) = (a/4, b/4), while this value decreases to 24.63 and 19.06 for the plate supported by the foundation with (a f , b f ) = (a/2, b/2) and (a f , b f ) = (3a/4, 3b/4), re- spectively. By comparing the frequency parameters in the three tables, one can see that the frequency parameter remarkably increases by increasing the foundation supporting area, regardless of the material grading index n and the foundation stiffness parame- ters. The effect of the shear deformation on the frequencies of the FGSW plate partially supported by the elastic foundation can also be seen from the tables, and the frequency
17. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 79 Table 4. Frequency parameter of SSSS square plates partially supported by elastic foundation with (a f , b f ) = (3a/4, 3b/4) a/h N Kw Ks (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-0-1) (1-8-1) 0 0 1.1886 1.2389 1.2867 1.3208 1.1343 1.5215 0.5 10 10 1.5946 1.6307 1.6672 1.6919 1.5578 1.8501 100 100 3.0190 3.0446 3.0713 3.0890 2.9940 3.0190 0 0 0.9637 1.0241 1.1006 1.1400 0.9159 1.4431 5 2 10 10 1.4497 1.4862 1.5404 1.5641 1.4269 1.7901 100 100 2.9412 2.9604 2.9974 3.0119 2.8567 3.1692 0 0 0.8934 0.9396 1.0267 1.0562 0.8678 1.4033 10 10 10 1.4204 1.4413 1.5001 1.5127 1.4176 1.7615 100 100 2.9568 2.9544 2.9904 2.9924 2.4105 3.1564 0 0 1.2590 1.3139 1.3688 1.4061 1.2033 1.2590 0.5 10 10 1.6646 1.7040 1.7465 1.7736 1.6273 1.6646 100 100 3.1780 3.2066 3.2402 3.2593 3.1541 3.1780 0 0 1.0096 1.0731 1.1586 1.1999 0.9666 1.5471 10 2 10 10 1.5022 1.5397 1.6009 1.6253 1.4862 1.8876 100 100 3.0758 3.0963 3.1445 3.1581 3.0847 3.3547 0 0 0.9351 0.9808 1.0773 1.1061 0.9295 1.4998 10 10 10 1.4720 1.4902 1.5558 1.5664 1.4914 1.8525 100 100 3.0900 3.0827 3.1318 3.1281 3.1446 3.3356 0 0 1.2861 1.3428 1.4007 1.4393 1.2298 1.6882 0.5 10 10 1.6916 1.7325 1.7775 1.8056 1.6542 2.0056 100 100 3.2514 3.2807 3.3175 3.3368 3.2300 3.4920 0 0 1.0268 1.0915 1.1806 1.2225 0.9860 1.5882 100 2 10 10 1.5219 1.5598 1.6238 1.6485 1.5090 1.9265 100 100 3.1377 3.1579 3.2116 3.2242 3.1563 3.4392 0 0 0.9507 0.9961 1.0963 1.1248 0.9538 1.5377 10 10 10 1.4914 1.5083 1.5768 1.5865 1.5212 1.8885 100 100 3.1517 3.1405 3.1960 3.1892 3.2354 3.4172 parameter increases by the increase of the side-to-thickness ratio. The numerical result in the tables shows the ability of the derived finite element formulation on modeling the shear deformation effect of the FGSW plate partially supported by the elastic foundation. The effect of the foundation support on the free vibration of the SSSS plate can be also seen from Fig.2 where the first vibration mode for the transverse displacement w of the plate is shown for various values of the foundation supporting areas. The first mode shape of the plate partially supported by the foundation, as seen from the figure is asymmetrical while that of the plate fully supported by the foundation is symmetrical.
18. 80 Le Cong Ich, Pham Vu Nam, Nguyen Dinh Kien The foundation supporting area is also plays an important role on the vibration mode of the plate, and the position at which the transverse bending displacement attains the maximum value depends on the foundation supporting area. (a) (a f , b f ) = (a/4, b/4) (b) (a f , b f ) = (a/2, b/2) a)a) (a(faa), fb, fb) (f(=)aa =ff,(, ab(b/4,aff))/4, == b (/4)(baa/4)/4,/4, bb/4)/4) b)b) ((aab)b)ff,, bbf(f))(a a==f,f , b( (baf)f/2,) = = ( ba(/2)a/2,/2, b b/2)/2) (c)c)(a f(,ab f, )b =) = (3 (3a/4,a/4, 3b /43b)/4) (d) (a f ,d)b f ) =(a (,a b, b)) = (a, b) c)c) (a(fa,c) fb, fb) f(=)a =ff(3, (3baff)/4,a =/4, 3(3 b3a/4)b/4,/4) 3b/4) d)d) ((aad)ff,, bbff))( a==f f ,( baf f,) b=) (a, b) FigureFigure Figure 2 .2 The. The 2first firstThe mode modefirstfirst shapesmodemode shapes shapesshapes for for transverse transverse forfor transversetransverse displacement displacement displacementdisplacement ofof SSSSSSSS of of SSSS SSSS(2(2 1 2)(2 (2 -square-11-2)-2) square square plate plate forforplate for for a/h=10, n=2, (Kw, Ks) = (50, 50) and different foundation supporting areas aFig./ah/=10,h 2=10,.a The /nh =2=10,n=2 first, (,K (nK modew=2,w K,, Ks()Ks )= shapesw = ,(50, K(50,s) =50) for50) (50, and transverse and 50) different different and different displacement foundation foundation foundation of supportingsupporting SSSS supporting (2-1-2) areas square areas plate for a/h = 10, n = 2, (Kw, Ks) = (50, 50) and different foundation supporting areas 4.34.3 Plate Plate4.3 withPlate with other otherwith boundary other boundary boundary conditions conditions conditionsconditions TheThe free freeThe vibration vibrationfree vibration characteristics characteristics characteristicscharacteristics of of the the ofofFGSW FGSW thethe FGSWFGSW square square squaresquare plateplate platewithplatewith other withotherwith other boundaryotherboundary boundaryboundary conditions conditions conditions (B.C.)(B.C.) (B.C.) (B.C.),, , , namelynamelynamely clamped clamped4.3. clamped Plate at attwo two with atopposite oppositetwo other oppositeopposite edges boundaryedges edgesandedges and simply simply conditionsandand simplysimply supported supported supportedsupported atat thethe at attwotwo thethe others others twotwo others others (CSCS)(CSCS) (CSCS) (CSCS) and clampedand and clamped clamped atat allall at at all all edges (CCCC), are reported in this subsection. In Table 5, the frequency parameters of the CSCS and edgesedges (CCCC)edges (CCCC) (CCCC), The are, are reportedfree , reported are vibration reported in in this this characteristics in su su thisbsection.bsection. subsection. In In Table of Table In the Table 5, 5, FGSW the the 5, frequency frequency squarethe frequency plate parameters parameters with parameters otherof the of boundary CSCS the CSCS and and and CCCCCCCC plates plates with witha/h=10 a/h are=10 listed are listed for ( aforf, b (f)a f=, b(fa) /2,= (ba/2)/2, andb/2) variousand various values values of the of foundation the foundation stiffness stiffness CCCC CCCCplatesconditions withplates a /withh=10 (B.C.), a are/h=10 namelylisted are forlisted clamped (a f,for bf) (=af at,( ab two/2,f) = b (/2) oppositea/2, and b/2) various and edges various values and values simplyof the offoundation supported the foundation stiffness at the stiffness parametersparameters and the and layer the layer thickness thickness ratio. ratio.As expected, As expected, the frequency the frequency parameter parameters of sthe of CCCCthe CCCC plate plate are are parametersparameterstwo and others the and layer (CSCS)the thickness layer and thickness clampedratio. As ratio. atexpected, allAs edges expected, the (CCCC), frequency the frequency are parameter reported parameters inof thisthes of CCCC subsection. the CCCC plate plateInare are higherhigher than than the corresponding the corresponding parameters parameters of the of CSCS the CSCS and SSSS and SSSS plates, plates, regardless regardless of the of foundation the foundation higher higher thanTab. the than5 , corresponding the the frequency corresponding parameters parameters parameters of of the the of CSCS CSCS the CSCS and and SSSS CCCC and SSSS plates, plates plates, regardless with regardlessa/ h of= the10 of arefoundation the listed foundation stiffnessstiffness and the and layer the layerthickness thickness ratio. ratio. The dependenceThe dependence of the of frequency the frequency parameter parameter of the of CSCS the CSCS and CCCCand CCCC stiffnessstif andfnessfor the( aand layer, b the) thickness =layer (a/2, thickness ratio.b/2) andThe ratio. dependence various The dependence values of the of offrequency the the foundation frequency parameter parameter stiffness of the of parametersCSCS the CSCS and CCCC and and CCCC platesplates upon uponthe fmaterial thef material grading grading index index and the and layer the layerthickness thickness ratio ratiois similar is similar to that to of that the of SSSS the SSSS plate. plate. plates uponplatesthe the upon layermaterial the thickness material grading grading ratio.index and Asindex expected,the and layer the thickness layer the frequencythickness ratio is ratio similar parameters is similar to that ofto of that the the CCCCofSSSS the SSSSplate. plate plate. are Figs. Figs.3 and 3 4 and respectively 4 respectively illustrate illustrate the first the fourfirst vibrationfour vibration modes modes for the for transverse the transverse displacement displacement w w Figs.higher 3 Figs.and 4 than3 respectively and the4 respectively corresponding illustrate illustrate theparameters first the four first vibration four of thevibration modes CSCS modes andfor the SSSS for transverse the plates, transverse displacement regardless displacement of w w of theof CSCS thethe CSCS foundationand CCCCand CCCC plates stiffness plates having and having a/ theh=10, a layer/h =10,n=2 thickness, npartially=2, partially supported ratio. supported The by dependence the by elastic the elastic foundation of thefoundation frequency with with(af, (af, of the CSCSof the andCSCS CCCC and CCCCplates havingplates having a/h=10, a /hn=10,=2, partially n=2, partially supported supported by the by elastic the elastic foundation foundation with (withaf, (af, bf) = b(af)/4, = (ba/4)/4, andb/4) (andKw, K(Ks)w ,= K (50,s) = 50)(50,. The50). influenceThe influence of the of foundation the foundation support support on the on vibration the vibration modes modes of of bf) = (ab/4,f) =b /4)(a/4, and b/4) (K wand, K s()K =w ,(50, Ks) =50) (50,. The 50) influence. The influence of the offoundation the foundation support support on the on vibration the vibration modes modes of of the platesthe plates can be can clearly be clearlyclearly seen seenfromseen fromfromthe figures. thethe figures.figures. The symmetry TheThe symmetrysymmetry of the ofof vibration thethe vibration vibration modes modes modes as seen as as seenfor seen the for for plate the the plate plate thefully plates fullysupported can supported be clearlyby the by foundationseen the foundation from theis destroyed figures. is destroyed Theby the symmetry by partial the partial foundation of the foundation vibration support. support. modes as seen for the plate fully supportedfully supported by the foundationby the foundation is destroyed is destroyed by the bypartial the partial foundation foundation support. support. 15 15 15 15
19. Free vibration of FG sandwich plates partially supported by elastic foundation using a quasi-3D finite element formulation 81 parameter of the CSCS and CCCC plates upon the material grading index and the layer thickness ratio is similar to that of the SSSS plate. Table 5. Frequency parameter of CSCS and CCCC square plates with a/h = 10, partially sup- ported by the foundation of (a f , b f ) = (a/2, b/2) B.C. NKw Ks (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-0-1) (1-8-1) 0 0 1.8227 1.9012 1.9701 2.0315 1.7409 2.3579 0.5 10 10 1.9922 2.0630 2.1262 2.1823 1.9199 2.4868 100 100 2.7998 2.8619 2.9177 2.9655 2.7356 3.2263 0 0 1.6163 1.7075 1.7947 1.8692 1.5332 2.2859 1 10 10 1.8098 1.8896 1.9681 2.0347 1.7399 2.4196 100 100 2.6359 2.7096 2.7818 2.8397 2.5691 3.1711 CSCS 0 0 1.4693 1.5614 1.6621 1.7422 1.4025 2.2289 2 10 10 1.6850 1.7627 1.8516 1.9216 1.6333 2.3669 100 100 2.5173 2.5939 2.6799 2.7422 2.4623 3.1287 0 0 1.3622 1.4302 1.5398 1.6101 1.3405 2.1636 10 10 10 1.6026 1.6550 1.7491 1.8076 1.5951 2.3072 100 100 2.4376 2.4929 2.5903 2.6434 2.4172 3.0820 0 0 2.2328 2.3278 2.4078 2.4839 2.1315 2.8698 0.5 10 10 2.3906 2.4781 2.5526 2.6234 2.2986 2.9884 100 100 3.2398 3.3118 3.3735 3.4313 3.1639 3.7334 0 0 1.9850 2.0960 2.1970 2.2907 1.8806 2.7851 1 10 10 2.1666 2.2663 2.3587 2.4445 2.0754 2.9082 100 100 3.0572 3.1404 3.2179 3.2878 2.9796 3.6690 CCCC 0 0 1.8076 1.9205 2.0365 2.1389 1.7208 2.7180 2 10 10 2.0117 2.1101 2.2143 2.3065 1.9403 2.8452 100 100 2.9295 3.0138 3.1032 3.1785 2.8649 3.6197 0 0 1.6769 1.7621 1.8873 1.9807 1.6362 2.6412 10 10 10 1.9064 1.9756 2.0852 2.1663 1.8796 2.7737 100 100 2.8478 2.9078 3.0052 3.0707 2.8090 3.5658 Figs.3 and4 respectively illustrate the first four vibration modes for the transverse displacement w of the CSCS and CCCC plates having a/h = 10, n = 2, partially sup- ported by the elastic foundation with (a f , b f ) = (a/4, b/4) and (Kw, Ks) = (50, 50). The influence of the foundation support on the vibration modes of the plates can be clearly seen from the figures. The symmetry of the vibration modes as seen for the plate fully supported by the foundation is destroyed by the partial foundation support.
20. Table 5: Frequency parameter of CSCS and CCCC square plates with a/h=10, partially supported by the foundation of (af, bf) = (a/2, b/2) B.C. N Kw Ks (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-0-1) (1-8-1) 0 0 1.8227 1.9012 1.9701 2.0315 1.7409 2.3579 0.5 10 10 1.9922 2.0630 2.1262 2.1823 1.9199 2.4868 100 100 2.7998 2.8619 2.9177 2.9655 2.7356 3.2263 0 0 1.6163 1.7075 1.7947 1.8692 1.5332 2.2859 1 10 10 1.8098 1.8896 1.9681 2.0347 1.7399 2.4196 100 100 2.6359 2.7096 2.7818 2.8397 2.5691 3.1711 CSCS 0 0 1.4693 1.5614 1.6621 1.7422 1.4025 2.2289 2 10 10 1.6850 1.7627 1.8516 1.9216 1.6333 2.3669 100 100 2.5173 2.5939 2.6799 2.7422 2.4623 3.1287 0 0 1.3622 1.4302 1.5398 1.6101 1.3405 2.1636 10 10 10 1.6026 1.6550 1.7491 1.8076 1.5951 2.3072 100 100 2.4376 2.4929 2.5903 2.6434 2.4172 3.0820 0 0 2.2328 2.3278 2.4078 2.4839 2.1315 2.8698 0.5 10 10 2.3906 2.4781 2.5526 2.6234 2.2986 2.9884 100 100 3.2398 3.3118 3.3735 3.4313 3.1639 3.7334 0 0 1.9850 2.0960 2.1970 2.2907 1.8806 2.7851 1 10 10 2.1666 2.2663 2.3587 2.4445 2.0754 2.9082 100 100 3.0572 3.1404 3.2179 3.2878 2.9796 3.6690 CCCC 0 0 1.8076 1.9205 2.0365 2.1389 1.7208 2.7180 2 10 10 2.0117 2.1101 2.2143 2.3065 1.9403 2.8452 100 100 2.9295 3.0138 3.1032 3.1785 2.8649 3.6197 0 0 1.6769 1.7621 1.8873 1.9807 1.6362 2.6412 8210 10 Le Cong10 Ich,1.9064 Pham Vu Nam,1.9756 Nguyen 2.0852 Dinh Kien 2.1663 1.8796 2.7737 100 100 2.8478 2.9078 3.0052 3.0707 2.8090 3.5658 Fig. 3. TheFigure first 3. four The first mode four shapes mode shapes for transverse for transverse displacement displacement of of CSCS CSCS (2-1-2)(2-1-2) square plate plate with a/h=10, n=2, (a , b ) = (a/4, b/4) and (K , K ) = (50, 50) with a/h = 10, n = 2, (a f ,f b ff) = (a/4, b/4) andw (Ks w, Ks) = (50, 50) 16 Fig. 4. The firstFigure four 4. The mode first shapes four mode for shapes transverse for transverse displacement displacement of CCCC of CCCC (2-1-2) (2-1-2) square square plate with with ah=10, n=2, (af, bf) = (a/4, b/4) and (Kw, Ks) = (50, 50) a/h = 10, n = 2, (a f , b f ) = (a/4, b/4) and (Kw, Ks) = (50, 50) 4.4 Plate with different side-to-thickness ratios The effect of the side-to-thickness ratio a/h on the frequency parameter of the FGSW plate is illustrated in Fig. 5 for the (1-1-1) SSSS and CCCC square plates with n=2, partially supported by the elastic foundation (af, bf) = (a/2, b/2). The frequency parameter, as seen from the figure, steadily increases by increasing the aspect ratio, and the increase is the most significant for a/h between 5 and 20. The foundation stiffness also plays an important role on the dependence of the frequency parameter on the aspect ratio, the increase of frequency parameter by increasing the aspect ratio is more significant when the plates are supported by the foundation with higher stiffness. The result in Fig. 5 shows again the ability of the finite element formulation derived in the present work in modeling the shear deformation effect of the FGSW plate. Figure 5. Frequency parameter versus side-to-thickness ratio of (1-1-1) SSSS and CCCC square FGSW plates partially supported by elastic foundation with (af, bf) = (a/2, b/2), (n=2) 17
21. FreeFigure vibration of 4 FG. The sandwich first plates four partially mode supported shapes by for elastic transverse foundation using displacement a quasi-3D finite of element CCCC formulation (2-1-2) 83square plate with ah=10, n=2, (af, bf) = (a/4, b/4) and (Kw, Ks) = (50, 50) 4.4. Plate with different side-to-thickness ratios 4.4 Plate with different side-to-thickness ratios The effect of the side-to-thickness ratio a/h on the frequency parameter of the FGSW plateThe is effect illustrated of the side in Fig.-to-5thickness for the ratio (1-1-1) a/h SSSS on the and frequency CCCC parameter square plates of the with FGSWn plate= 2, is illustrated inpartially Fig. 5 for supported the (1-1- by1) theSSSS elastic and foundation CCCC square(a f , platesb f ) = with (a/2, n=2,b/2 )partially. The frequency supported pa- by the elastic foundationrameter, as(af seen, bf) from = (a/2, the b/2) figure,. The steadily frequency increases parameter, by increasing as seen from the aspect the figure, ratio, steadily and the increases by increasingincrease the is the aspect most ratio, significant and the increase for a/h isbetween the most 5 significant and 20. The for foundationa/h between stiffness 5 and 20. also The foundation stiffnessplays analso important plays an important role on the role dependence on the dependence of the frequency of the frequen parametercy parameter on the aspecton the aspect ra- ratio, the increasetio, the of increase frequency of frequency parameter parameter by increasing by increasing the aspect the ratio aspect is more ratio significant is more significant when the plates are supportedwhen the by plates the foundation are supported with byhigher the foundationstiffness. The with result higher in Fig. stiffness. 5 shows The again result the in ability Fig.5 of the finite elemshowsent formulation again the ability derived of in the the finite present element work formulation in modeling derived the shear in deformation the present workeffect inof the FGSW plate.modeling the shear deformation effect of the FGSW plate. Fig. 5.Figure Frequency 5. Frequency parameter parameter versus side-to-thickness versus side-to-thickness ratio of (1-1-1) ratio of SSSS (1-1 and-1) SSSS CCCC and square CCCC square FGSWFGSW plates plates partially partially supported supported by elastic by foundationelastic foundation with (a f with, b f ) =(a (f,a b/2,f) =b/2 (a)/,2(,n b=/2)2,) (n=2) 5. CONCLUSIONS The free vibration of FGSW plates partially supported by a Pasternak foundation has been studied using a quasi-3D finite element formulation.17 The plates are considered to be composed of three layers, a homogeneous ceramic core and two functionally graded skin layers. Mori–Tanaka scheme was employed to estimate the effective material properties of the plates. The frequency parameters and vibration modes have been evaluated for the FGSW plates with various boundary conditions, supported by the foundation of different areas. The numerical results obtained in the present paper reveal that the foundation supporting area plays an important role on both the frequencies and mode shapes of the plates. A parametric study has been carried out to highlight the influence of the material grading index, the layer thickness ratio and the foundation stiffness on the vibration characteristics of the plates. The effect of the side-to-thickness ratio on the frequencies of the FGSW plates has also been examined and discussed.
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