Applied Mathematics and Mechanics Free vibration of

Appl. Math. Mech. -Engl. Ed., 32(7), 925–942 (2011) DOI 10.1007/s10483-011-1470-9 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Applied Mathematics and Mechanics (English Edition)

Free vibration of functionally graded sandwich plates using four-variable refined plate theory∗ L. HADJI1,2 , H. A. ATMANE1,3 , A. TOUNSI1 , I. MECHAB1 , E. A. ADDA BEDIA1 (1. Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbes, Sidi Bel Abbes 22000, Algérie; 2. Département de Génie Civil, Université Ibn Khaldoun, Tiaret 14000, Algérie; 3. Département de Génie Civil, Faculté des Sciences de l’Ingénieur, Univesité Hassiba Benbouali de Chlef, Chlef 02000, Algérie) (Communicated by Ling-hui HE)

Abstract This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton’s principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-order theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates. Key words functionally graded material (FGM), free vibration, sandwich plate, refined plate theory (RPT), Navier solution Chinese Library Classification O354.1 2010 Mathematics Subject Classification

1

76G25

Introduction

The sandwich construction has been developed and utilized for almost 50 years because of its outstanding bending rigidity, low specific weight, superior isolating qualities, excellent vibration ∗ Received Oct. 18, 2010 / Revised Apr. 11, 2011 Corresponding author A. TOUNSI, Professor, E-mail: tou [email protected]

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L. HADJI, H. A. ATMANE, A. TOUNSI, I. MECHAB, and E. A. ADDA BEDIA

characteristics, and good anti fatigue properties. The sandwich composite construction offers great potential for large civil infrastructure projects, such as industrial buildings and vehicular bridges. The sandwich structure represents a special form of a layered structure that consists of two thin stiff and strong face sheets separated and a relatively thick, lightweight, and compliant core material. In modern sandwich structures, the faces are usually made of metal or laminated composite materials, and typically a compliant compressible core is made of a low-strength honeycomb type material or polymeric foam. The faces and the core are joined by adhesive bonding, which ensures the load transfer between the sandwich constituent parts. By the mid-1960s, the sandwich construction has been widely studied. For an extensive review of literature for the analysis of the sandwich structures, the reader may consult [1–5]. The methods of analyzing sandwich structures and numerical solutions for the standard problems are well documented in the books of Plantema[1] and Allen[2] . The structural analyses of constant-thickness sandwich composite structures are discussed in books of Whitney[3] and Vinson[5] , where they emphasized the importance of including the shear flexibility of the core. Pagano[6] and Pagano and Hatfield[7] presented the exact 3D elasticity solutions for the stress analysis of laminated composite and sandwich plates which serve as the benchmark solutions for comparison by many researchers. Moreover, the functionally graded materials (FGMs)[8–9] , a new generation of advanced inhomogeneous composite materials first proposed for thermal barriers[10], have been increasingly applied for modern engineering structures in the extremely high temperature environment. Many studies were conducted concerning the thermal mechanical behavior of FGMs[11–12] . In the simplest FGMs, two different material ingredients change gradually from one to the other. Discontinuous changes, such as a stepwise gradation of the material ingredients, can also be considered as an FGM. The most familiar FGM is compositionally graded from a refractory ceramic to a metal. Typically, FGMs are made from a mixture of ceramic and metal or a combination of different materials. The ceramic in an FGM offers thermal barrier effects and protects the metal from corrosion and oxidation, and the FGM is toughened and strengthened by the metallic composition. FGMs are now developed for general use as structural elements in extremely high temperature environments and different applications. Because of the wide application of FGMs, many studies have been performed to analyze the behavior and understand the mechanics and mechanism of FGM structures. Extensive studies have been carried out both theoretically and experimentally on fracture mechanics[13–14] , thermal stress distribution[15–17] , processing[18–19] and so on. Among these FGM structures, the plates and shells are still the interests for researchers because of their applications. Approaches, such as using the shear deformation plate theory, the energy method, and the finite-element method, have been carried out. Reddy[20] presented the solutions of the static behavior for the FGM rectangular plates based on his third-order shear deformation plate theory. Cheng and Batra[21] presented the results for the buckling and steady state vibrations of a simply supported FGM polygonal plate based on Reddy’s plate theory. Loy et al.[22] presented the Rayleigh-Ritz solutions for free vibration of simply supported cylindrical shells made of an FGM compound of stainless steel and nickel by using Love’s shell theory. Praveen and Reddy[23] investigated the nonlinear static and dynamic response of functionally graded ceramic-metal plates using a plate finite element that accounts for the transverse shear strains, rotary inertia, and moderately large rotations in the von Karman sense. The FGM sandwich can alleviate the large interfacial shear stress concentration because of the gradual variation of material properties at the facesheet-core interface. The effects of the FGM core were studied by Venkataraman and Sankar[24] and Anderson[25] on the shear stresses at the facesheet-core of the FGM sandwich beam. Pan and Han[26] analyzed the static response of the multilayered rectangular plate made of the functionally graded, anisotropic, and linear magneto-electro-elastic materials. Das et al.[27] studied a sandwich composed of a single FGM softcore with relatively orthotropic stiff facesheets by using a triangular plate element. Shen[28]

Free vibration of functionally graded sandwich plates

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considered two types of FGM hybrid laminated plates, one is with the FGM core and the piezoelectric ceramic facesheet, and the other is with the FGM facesheet and the piezoelectric ceramic core. The FGM sandwich construction exists in two types: the FGM facesheet-homogeneous core and the homogeneous facesheet-FGM core. For the case of the homogeneous core, the softcore is commonly employed because of the light weight and high bending stiffness in the structural design. The homogeneous hardcore is also employed in other fields, such as control or thermal environments. The actuators and sensors which are common piezoelectric ceramics are always in the midlayers of the sandwich construction (see [28]). Moreover, in the thermal environments, the metal-rich facesheet can reduce the large tensile stress on the surface at the early stage of cooling[29] . In general, the plates made of FGMs are not materially symmetric about the midplane for the special material properties distribution. Their stretching and flexural deformation modes are coupled. Recently, Li et al.[30] presented a three-dimensional solution for free vibration of multi-layer FGM system-symmetric and unsymmetric FGM sandwich plates using the Ritz method. As far as we know, there has been no investigation on free vibration of the FGM sandwich plates using the four-variable refined plate theory (RPT). This theory was developed by Shimpi[31] for isotropic plates and extended by Shimpi and Patel[32–33] for orthotropic plates. This theory which looks like a higher-order theory uses only two unknown functions in order to derive two governing equations for orthotropic plates. The most interesting feature of this theory is that it does not require the shear correction factor, and has strong similarities with the classical plate theory (CPT) in some aspects, such as governing equation, boundary conditions, and moment expressions. Lee et al.[34] proposed a higher-order shear deformable theory using the similar approach of representing transverse displacement using two components. Recently, Mechab et al.[35] developed the two variable RPT for FGM plates. The accuracy of this theory has been demonstrated for static bending of FGM plates by Mechab et al.[35] . Therefore, it seems to be important to extend this theory to the free vibration behaviors of FGM sandwich plates. Two common types of FGM sandwich plates, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core, are considered. The present theory satisfies equilibrium conditions at the top and bottom faces of the sandwich plate without using shear correction factors. The Navier solution is used to obtain the closed form solutions for simply supported FGM sandwich plates. To illustrate the accuracy of the present theory, the obtained results are compared with threedimensional elasticity solutions and the results of the first-order and the other higher-order theories.

2

RPT for FGM sandwich plates

2.1 Geometrical configuration Consider the case of a rectangular FGM sandwich plate with the uniform thickness composed of three microscopically heterogeneous layers referring to rectangular coordinates (x, y, z) as shown in Fig. 1. The top and bottom faces of the plate are at z = ±h/2, and the edges of the plate are parallel to axes x and y. The sandwich plate is composed of three elastic layers, namely, Layer 1, Layer 2, and Layer 3 from bottom to top of the plate. The vertical ordinates of the bottom, the two interfaces, and the top are denoted by h1 = −h/2, h2 , h3 , and h4 = h/2, respectively. For brevity, the ratio of the thickness of each layer from bottom to top is denoted by the combination of three numbers, i.e., “1-0-1”, “2-1-2” and so on. As shown in Fig. 2, Types A and B are considered in the present study, i.e., the FGM facesheet and the homogeneous core and the homogeneous facesheet and the FGM core.

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Fig. 1


Geometry of the rectangular FGM sandwich plate with uniform thickness in rectangular Cartesian coordinates

Fig. 2

Material variation along thickness of FGM sandwich plate

2.2 Material properties The properties of FGM vary continuously due to the gradually changing volume fraction of the constituent materials (ceramic and metal), usually in the thickness direction only. The power-law function is commonly used to describe these variations of materials properties. The sandwich structures made of two types of power-law FGMs mentioned before are discussed as follows. 2.2.1 Type A: power-law FGM facesheet and homogeneous core The volume fraction of the FGMs is assumed to obey a power-law function along the thickness direction z − h k 1 , z ∈ [h1 , h2 ], (1a) V (1) = h2 − h1 V (2) = 1, V (3) =

z ∈ [h2 , h3 ],

z − h k 4 , h3 − h4

z ∈ [h3 , h4 ],

(1b) (1c)

where V (n) (n = 1, 2, 3) denotes the volume fraction function of Layer n, and k is the volume fraction index (0 k +∞), which dictates the material variation profile through the thickness. 2.2.2 Type B: homogeneous facesheet and power-law FGM core The volume fraction of the FGMs is assumed to obey a power-law function along the thickness direction V (1) = 0, V (2) =

z ∈ [h1 , h2 ],

z − h k 2 , h3 − h2

V (3) = 1,

z ∈ [h2 , h3 ],

z ∈ [h3 , h4 ],

(2a) (2b) (2c)


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in which V (n) and k are defined as same as in Eq. (1). The effective material properties, i.e., Young’s modulus E, Poisson’s ratio ν, and the mass density ρ, can be expressed by the rule of mixture[14] as P (n) (z) = P2 + (P1 − P2 )V (n) ,

(3)

where P (n) is the effective material property of the FGM of Layer n. For Type A, P1 and P2 are the properties of the top and bottom faces of Layer 1, respectively, and vice versa for Layer 3 depending on the volume fraction V (n) (n = 1, 2, 3). For Type B, P1 and P2 are the properties of Layer 3 and Layer 1, respectively. These two types of FGM sandwich plates will be discussed later in the following sections. For simplicity, Poisson’s ratio of the plate is assumed to be constant in this study as the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus[36] . 2.3 Basic assumptions Assumptions of the present RPT are as follows: (i) The displacements are small in comparison with the plate thickness. Therefore, the strains involved are infinitesimal. (ii) The transverse displacement W includes two components: bending displacement wb and shear displacement ws . These components are the functions of coordinates x, y, and time t only. W (x, y, z, t) = wb (x, y, t) + ws (x, y, t).

(4)

(iii) The transverse normal stress σz is negligible in comparison with the in-plane stresses σx and σy . (iv) The displacements U in the x-direction and V in the y-direction consist of extension, bending, and shear components. U = u + ub + us ,

V = v + vb + vs .

(5)

The bending components ub and vb are assumed to be similar to the displacements given by the classical plate theory. Therefore, the expression for ub and vb can be given as ub = −z

∂wb , ∂x

vb = −z

∂wb . ∂y

(6a)

The shear components us and vs give rise, in conjunction with ws , to the parabolic variations of shear strains γxz , γyz and hence to shear stresses τxz , τyz through the thickness of the plate in such a way that shear stresses τxz , τyz are zero at the top and bottom faces of the plate. Consequently, the expression for us and vs can be given as 1 1 5 z 2 ∂ws 5 z 2 ∂ws us = z− z , vs = z− z . (6b) 4 3 h ∂x 4 3 h ∂y 2.4 Kinematics and constitutive equations Based on the assumptions made in the preceding section, the displacement field can be obtained using Eqs. (4)–(6) as ⎧ 1 5 z 2 ∂w ∂wb s ⎪ U (x, y, z) = u(x, y) − z ⎪ + z − , ⎪ ⎪ ∂x 4 3 h ∂x ⎪ ⎨ 1 5 z 2 ∂w ∂wb s (7) + z − , V (x, y, z) = v(x, y) − z ⎪ ⎪ ∂y 4 3 h ∂y ⎪ ⎪ ⎪ ⎩ W (x, y, z) = wb (x, y) + ws (x, y).

930


The strains associated with the displacements in Eq. (7) are ⎧ εx = ε0x + zkxb + f kxs , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ εy = ε0y + zkyb + f kys , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 b s ⎨ γxy = γxy + zkxy + f kxy ,

(8)

⎪ s ⎪ , γyz = gγyz ⎪ ⎪ ⎪ ⎪ ⎪ s ⎪ ⎪ ⎪ γxz = gγxz , ⎪ ⎪ ⎪ ⎩ εz = 0, where ⎧ ∂u ∂ 2 wb ∂ 2 ws ⎪ ⎪ ε0x = , kxb = − , kxs = − , ⎪ ⎪ 2 ⎪ ∂x ∂x ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂v ∂ 2 wb ∂ 2 ws ⎪ 0 b s ⎪ , k = = − , k = − , ε ⎪ y y y ⎪ ⎪ ∂y ∂y 2 ∂y 2 ⎪ ⎪ ⎨ ∂u ∂v ∂ 2 wb ∂ 2 ws 0 b s + , kxy , kxy , = = −2 = −2 γxy ⎪ ⎪ ∂y ∂x ∂x∂y ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂ws ∂ws ⎪ s s ⎪ γyz , γxz , = = ⎪ ⎪ ∂y ∂x ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ f = −1z + 5z z , g = 5 − 5 z . 4 3 h 4 h

(9)

For elastic and isotropic FGMs, the constitutive relations can be written as ⎤ ⎡ Q11 σx ⎣ σy ⎦ = ⎣ Q12 0 τxy ⎡

Q12 Q22 0

⎤⎡ ⎤ 0 εx 0 ⎦ ⎣ εy ⎦ , Q66 γxy

τyz τzx

=

Q44 0

0 Q55 l

γyz γzx

,

(10)

where (σx , σy , τxy , τyz , τzx ) and (εx , εy , γxy , γyz , γzx ) are the stress and strain components, respectively. Using the material properties defined in Eq. (3), stiffness coefficients Qij , can be expressed as Q11 = Q22 = Q12 =

E(z) , 1 − ν2

νE(z) , 1 − ν2

Q44 = Q55 = Q66 =

(11a) (11b)

E(z) . 2 (1 + ν)

(11c)

2.5 Governing equations The strain energy of the plate can be written as 1 Ue = 2

V

(σx εx + σy εy + τxy γxy + τyz γyz + τzx γzx )dV.

(12)


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By substituting Eqs. (8) and (10) into Eq. (12) and integrating through the thickness of the sandwich plate, the strain energy of the plate can be rewritten as

Ue =

1 2

A

b b kxy + Mxs kxs Nx ε0x + Ny ε0y + Nxy ε0xy + Mxb kxb + Myb kyb + Mxy

s s s s s s dxdy, +Mys kys + Mxy kxy + Syz γyz + Sxz γxz

(13)

where the stress resultants N , M , and S are defined by ⎧ 3 hn+1 ⎪ ⎪ ⎪ ⎪ (N , N , N ) = (σx , σy , τxy ) dz, x y xy ⎪ ⎪ ⎪ hn ⎪ n=1 ⎪ ⎪ ⎪ ⎪ ⎪ 3 hn+1 ⎪ ⎪ b ⎪ b b ⎪ (σx , σy , τxy ) zdz, Mx , My , Mxy = ⎪ ⎪ ⎪ ⎨ hn n=1

(14)

⎪ 3 hn+1 ⎪ ⎪ ⎪ s ⎪ Mxs , Mys , Mxy = (σx , σy , τxy ) f dz, ⎪ ⎪ ⎪ ⎪ n=1 hn ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 hn+1 ⎪ s ⎪ ⎪ s ⎪ S = , S (τxz , τyz ) gdz. ⎪ ⎩ xz yz n=1

hn

Substituting Eq. (10) into Eq. (14) and integrating through the thickness of the plate, the stress resultants are given as ⎡

⎤ ⎡ N A ⎣ Mb ⎦ = ⎣ B Ms Bs

B D Ds

⎤⎡ ⎤ ε Bs Ds ⎦ ⎣ kb ⎦ , Hs ks

s Syz s Sxz

=

As44 0

0 As55

s γyz s γxz

,

(15)

where T

N = (Nx , Ny , Nxy ) , 0 ε = ε0x , ε0y , γxy ,

b T M b = Mxb , Myb , Mxy ,

b k b = kxb , kyb , kxy ,

⎡

⎤ A11 A12 0 A = ⎣ A12 A22 0 ⎦ , 0 0 A66 ⎤ s s B12 0 B11 s s B22 0 ⎦, B s = ⎣ B12 s 0 0 B66 ⎡

s T M s = Mxs , Mys , Mxy ,

s k s = kxs , kys , kxy ,

⎡

⎤ B11 B12 0 B = ⎣ B12 B22 0 ⎦ , 0 0 B66 ⎤ s s D12 0 D11 s s D22 0 ⎦, Ds = ⎣ D12 s 0 0 D66 ⎡

(16a) (16b)

⎡

⎤ D11 D12 0 D = ⎣ D12 D22 0 ⎦ , 0 0 D66 ⎤ s s H12 0 H11 s s H22 0 ⎦, H s = ⎣ H12 s 0 0 H66

(16c)

⎡

(16d)

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where ⎧ 3 hn+1 ⎪ ⎪ ⎪ (A 1, z, z 2, z 3 , z 4 , z 6 Qij dz, , B , D , E , F , H ) = ij ij ij ij ij ij ⎪ ⎪ ⎪ ⎪ n=1 hn ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 5 ⎪ s ⎪ Bij = − Bij + 2 Eij , i, j = 1, 2, 6, ⎪ ⎪ 4 3h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 5 ⎪ ⎪ Ds = − Dij + 2 Fij , i, j = 1, 2, 6, ⎪ ⎪ ⎨ ij 4 3h

i, j = 1, 2, 6,

⎪ 1 5 25 ⎪ ⎪ s ⎪ ⎪ Hij = 16 Dij − 6h2 Fij + 9h4 Hij , i, j = 1, 2, 6, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 hn+1 ⎪ ⎪ 2 4 ⎪ ⎪ ⎪ (Aij , Dij , Fij ) = 1, z , z Qij dz, i, j = 4, 5, ⎪ ⎪ ⎪ ⎪ n=1 hn ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ As = 25 A − 25 D + 25 F , i, j = 4, 5. ij ij ij ij 16 6h2 h4

(17)

The kinetic energy of the plate can be written as 1 T = 2 =

1 2

¨ 2 + V¨ 2 + W ¨ 2 )dV ρ(U V

+

2

A

1 2

I0 (¨ u2 + v¨2 + (w ¨b + w ¨s ) )dxdy ∂w ¨b 2 ∂ w ¨s 2 ∂ w ¨b 2 I2 ∂ w ¨s 2 I2 + dxdy, + + ∂x ∂y 84 ∂x ∂y A

(18)

where ρ is the mass of density of the FG plate, and Ii (i = 0,2) are the inertias defined by

(I0 , I2 ) =

3 n=1

hn+1

(1, z 2 )ρdz.

(19)

hn

Hamilton’s principle[36] is used to derive the equations of motion appropriate to the displacement field and the constitutive equation. The principle can be stated in an analytical form as 0= 0

t

δ(Ue − T )dt,

(20)

where δ indicates a variation with respect to x and y. By substituting Eqs. (13) and (18) into Eq. (20) and integrating the equation by parts and collecting the coefficients of δu, δv, δwb , and δws , the equations of motion for the FG sandwich


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plate are obtained as follows: ⎧ ⎪ ⎪ δu : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δv : ⎪ ⎪ ⎨

∂Nx ∂Nxy + = I0 u ¨, ∂x ∂y ∂Ny ∂Nxy + = I0 v¨, ∂x ∂y

b ∂2w ⎪ ∂ 2 Mxy ∂ 2 Myb ∂ 2 Mxb ∂2w ¨b ¨b ⎪ ⎪ ⎪ δw : + 2 = I ( w ¨ + w ¨ ) − I + , + b 0 b s 2 ⎪ ⎪ ∂x2 ∂x∂y ∂y 2 ∂x2 ∂y 2 ⎪ ⎪ ⎪ ⎪ ⎪ s s s ⎪ ∂ 2 Mys ∂Sxz ∂Syz ∂ 2Mxy ⎪ ¨s ∂ 2 w ¨s ∂ 2 Mxs I2 ∂ 2 w ⎪ ⎩ δws : . + + = I0 (w¨b + w +2 + ¨s ) − + 2 2 2 ∂x ∂x∂y ∂y ∂x ∂y 84 ∂x ∂y 2

(21)

Equation (21) can be expressed in terms of displacements (u, v, wb , ws ) by substituting the stress resultants from Eq. (15). For FG plates, the equilibrium equations (21) take the forms

A11

∂2u ∂2u ∂ 3 wb ∂2v ∂ 3 wb − B + A + (A + A ) − (B + 2B ) 66 12 66 11 12 66 ∂x2 ∂y 2 ∂x∂y ∂x3 ∂x∂y 2

s − B11

∂ 3 ws ∂ 3 ws s s − (B12 + 2B66 ) = I0 u ¨, 3 ∂x ∂x∂y 2

(A12 + A66 ) s − B22

B11

∂2u ∂ 3 wb ∂2v ∂2v ∂ 3 wb + A66 2 + A22 2 − (B12 + 2B66 ) 2 − B22 ∂x∂y ∂x ∂y ∂x ∂y ∂y 3

∂ 3 ws ∂ 3 ws s s = I0 v¨, − (B + 2B ) 12 66 ∂y 3 ∂x2 ∂y

s − D22

4 ∂ 4 wb ∂ 4 wb ∂ 4 ws s ∂ ws s s − D − D − 2 (D + 2D ) 22 11 12 66 ∂x2 ∂y 2 ∂y 4 ∂x4 ∂x2 ∂y 2

∂ 4 ws = I0 (w ¨b + w ¨s ) − I2 ∇2 w ¨b , ∂y 4

(22c)

3 4 ∂3u ∂3v ∂3u s s s s s ∂ v s ∂ wb + B22 + (B12 + 2B66 ) + (B12 + 2B66 ) 2 − D11 3 2 3 ∂x ∂x∂y ∂x ∂y ∂y ∂x4

s s − 2 (D12 + 2D66 ) s − H22

2.6

(22b)

∂3u ∂3v ∂ 4 wb ∂3u ∂3v + B22 3 − D11 + (B12 + 2B66 ) + (B12 + 2B66 ) 2 3 2 ∂x ∂x∂y ∂x ∂y ∂y ∂x4

− 2 (D12 + 2D66 )

s B11

(22a)

4 4 ∂ 4 wb ∂ 4 ws s ∂ wb s ∂ ws s s − D − H − 2 (H + 2H ) 22 11 12 66 ∂x2 ∂y 2 ∂y 4 ∂x4 ∂x2 ∂y 2

2 2 ∂ 4 ws I2 s ∂ ws s ∂ ws + A + A = I0 (w ¨b + w ¨s ) − ∇2 w ¨b . 55 44 ∂y 4 ∂x2 ∂y 2 84

(22d)

Navier solution for simply supported rectangular sandwich plates

Rectangular sandwich plates are generally classified in accordance with the used type support. We are here concerned with the analytical solutions of Eq. (22) for the simply supported

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FG sandwich plate. The following boundary conditions are imposed at the side edges. a a ∂w a ∂w a a b s − ,y = − , y = 0, v − , y = wb − , y = ws − , y = 2 2 2 ∂y 2 ∂y 2 v

a

a a ∂w a ∂w a b s , y = wb , y = ws , y = ,y = , y = 0, 2 2 2 ∂y 2 ∂y 2

(23a)

(23b)

a a a a a a Nx − , y = Mxb − , y = Mxs − , y = Nx , y = Mxb , y = Mxs , y = 0, (23c) 2 2 2 2 2 2 b b ∂wb b b ∂ws b u x, − = wb x, − = ws x, − = x, − = x, − = 0, 2 2 2 ∂x 2 ∂x 2

(23d)

b b ∂w b ∂w b b b s u x, = wb x, = ws x, = x, = x, = 0, 2 2 2 ∂x 2 ∂x 2

(23e)

b b b b b b Ny x, − = Myb x, − = Mys x, − = Ny x, = Myb x, = Mys x, = 0. 2 2 2 2 2 2

(23f)

The displacement functions that satisfy the equations of boundary conditions (23) are selected as the following Fourier series: ⎡ ⎤ u ∞ ∞ ⎢ ⎢ v ⎥ ⎢ ⎢ ⎥= ⎢ ⎣ wb ⎦ ⎣ m=1 n=1 ws ⎡

Umn cos(λx) sin(μy)eiωt Vmn sin(λx) cos(μy)eiωt Wbmn sin(λx) sin(μy)eiωt Wsmn sin(λx) sin(μy)eiωt

⎤ ⎥ ⎥ ⎥, ⎦

(24)

where Umn , Vmn , Wbmn , and Wsmn are arbitrary parameters to be determined, ω is the eigenfrequency associated with the (m,n)th eigenmode, and λ = mπ/a and μ = nπ/b. Substituting Eqs. (17), (19) and (24) into the equations of motion (22), we get the below eigenvalue equations for any fixed values of m and n for the free vibration problem K − ω 2 M Δ = 0,

(25)

ΔT = (Umn , Vmn , Wbmn , Wsmn ),

(26)

where Δ denotes the column

and ⎡

a11 ⎢ a12 K=⎢ ⎣ a13 a14

a12 a22 a23 a24

a13 a23 a33 a34

⎤ a14 a24 ⎥ ⎥, a34 ⎦ a44

⎡

m11 ⎢ 0 M =⎢ ⎣ 0 0

0 m22 0 0

0 0 m33 m34

⎤ 0 0 ⎥ ⎥, m34 ⎦ m44

(27)


935

in which ⎧ a11 = A11 λ2 + A66 μ2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a12 = λμ (A12 + A66 ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a13 = −λ(B11 λ2 + (B12 + 2B66 )μ2 ), ⎪ ⎪ ⎪ ⎪ s s s ⎪ ⎪ a14 = −λ(B11 λ2 + (B12 + 2B66 )μ2 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a22 = A66 λ2 + A22 μ2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a23 = −μ((B12 + 2B66 )λ2 + B22 μ2 ), ⎪ ⎪ ⎪ ⎪ ⎨ s s s + 2B66 )λ2 + B22 μ2 ), a24 = −μ((B12 ⎪ ⎪ ⎪ a33 = D11 λ4 + 2(D12 + 2D66 )λ2 μ2 + D22 μ4 , ⎪ ⎪ ⎪ ⎪ ⎪ s s s s ⎪ a34 = D11 λ4 + 2(D12 + 2D66 )λ2 μ2 + D22 μ4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s 4 s s s 4 ⎪ λ + 2(H12 + 2H66 )λ2 μ2 + H22 μ + As55 λ2 + As44 μ2 , a44 = H11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m11 = m22 = m34 = I0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m33 = I0 + I2 λ2 + μ2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ m44 = I0 + I2 λ2 + μ2 . 84

(28)

2.7

Ritz solution of vibration problem of rectangular sandwich plates with various boundary conditions The Ritz method is a variational approach and requires the expansion of the unknown functions of displacement components in an infinite series form. By taking a sufficient number of terms in these series, it is possible to approach the exact solution of the problem considered. However, the displacement functions should be complete in the space of functions, and inappropriate choices of the unknown functions can cause a very slow converging rate and numerical instabilities. The functions chosen by the researchers are trigonometric functions[38] , algebraic polynomials[39–41] , and orthogonal polynomials[42–43] employed on the basis of different plate theories. After defining the nondimensional coordinates as ξ = 2x/a and η = 2y/b choosing the origin of the coordinates as −1 ξ 1 and −1 η 1, we assume the displacement components as following simple algebraic polynomials which are the power functions of coordinate parameters in the expansion of double infinite series: u(ξ, η, t) =

I−1 J−1

Aij Xi (ξ)Yj (η) sin(ωt),

(29a)

i=0 j=0

v(ξ, η, t) =

K−1 L−1

Bkl Xk (ξ)Yl (η) sin(ωt),

(29b)

k=0 l=0

wb (ξ, η, t) =

M−1 −1 N

Cmn Xm (ξ)Yn (η) sin(ωt),

(29c)

m=0 n=0

ws (ξ, η, t) =

P −1 Q−1 p=0 q=0

Dpq Xp (ξ)Yq (η) sin(ωt),

(29d)

936


where the polynomials are defined as Xf (ξ) = ξ f (ξ + 1)

B1

(ξ − 1)

B3

,

Yg (ξ) = η g (η + 1)B2 (η − 1)B4 ,

f = i, k, m, p,

(30a)

g = j, l, n, q,

(30b)

and Aij , Bkl , Cmn and Dpq are unknown constant coefficients. Here, Bi can take the values which are chosen according to the type of boundary conditions imposed at the edges of the plate as the i subindices of Bi denote the numbers of the subsequent edges of the plate in the counterclockwise direction. The edge numbered as 1 is the one at ξ = −1. The values of Bi that are 0, 1 and 2 correspond to the free, simply supported, and clamped edges, respectively[44] . By keeping in mind that the Ritz method satisfies only the geometric boundary conditions, it is possible to satisfy any set of geometric edge boundary conditions. For the Ritz method, the natural boundary conditions for clamped plates are required. The boundary conditions of a plate (of length a and width b) are given as follows: Clamped-clamped boundaries u = v = wb = ws =

∂ws ∂wb = =0 ∂η ∂η

at ξ = ±1,

(31a)

u = v = wb = ws =

∂ws ∂wb = =0 ∂ξ ∂ξ

at η = ±1.

(31b)

Free-free boundaries (no constraints) u = 0,

v = 0,

wb = 0,

ws = 0,

u = 0,

v = 0,

wb = 0,

ws = 0,

∂wb = 0, ∂η ∂wb = 0, ∂ξ

∂ws = 0 at ξ = ±1, ∂η ∂ws = 0 at η = ±1. ∂ξ

(31c) (31d)

Inserting the displacement forms (Eq. (29)) into the kinetic and strain energy definitions (Eqs. (12) and (18)) and minimizing the Lagrangian of the system with respect to the coefficients of the displacement functions for the vibration problem yield the algebraic simultaneous equations with the same number of unknown coefficients given in Eq. (29). The number of these equations becomes 5M 2 if the same number of terms, say M , is employed in all of the series for convenience. The algebraic equations obtained will be given in the form of the generalized eigenvalue problem (Eq. (25)). For a non-trivial solution, the eigenvalues (ω), which make the determinant to be equal to zero, correspond to the free vibration frequencies.

3

Numerical results and discussion

In this study, the free vibration analysis of simply supported FG sandwich plates by the present RPT is suggested for investigation. Navier solutions for the free vibration analysis of FG sandwich plates are presented by solving the eigenvalue equations. The FG plate is taken to be made of aluminum and alumina with the following material properties: Ceramic (P1 , alumina, Al2 O3 ) Ec = 380 GPa, ν = 0.3, and ρc = 3 800 kg/m3 . Metal (P2 , aluminium, Al) Em = 70 GPa, ν = 0.3, and ρm = 2 707 kg/m3 . For simplicity, the nondimensional natural frequency parameter is defined as ωb2 ρ0 = , (32) h E0 where ρ0 = 1 kg/m3 , and E0 = 1 GPa.


937

Various numerical examples are described and discussed for verifying the accuracy of RPT in predicting the free vibration behaviors of simply supported FG sandwich plates. For the verification purpose, the results obtained by the present RPT are compared with other theories existing in the literature, such as the classical plate theory (CPT), the first-order shear deformation plate theory (FSDPT), the third-order shear deformation plate theory (TSDPT), and the sinusoidal shear deformation plate theory (SSDPT). We also take the shear correction factor K = 5/6 in FSDPT. The results of the power-law FGM sandwich plates of Type A with six material distributions are compared in Table 1 with the results of CPT, FSDPT, TSDPT, SSDPT, and the threedimensional linear theory of elasticity[30] . Young’s modulus E and the mass density ρ are based on the power-law distribution Eq. (3). Table 1 shows a good agreement by the comparisons of FGM plates of five different volume fraction indices k = 0, 0.5, 1, 5, 10 with other theories. In general, the vibration frequencies obtained by CPT are much higher than those computed from the shear deformation theories. This implies the well-known fact that the results estimated by CPT are grossly in error for a thick plate. Comparisons are given in Tables 2 and 3 on the basis of the homogeneous hardcore and homogeneous softcore types of FG sandwich plates (Type A). Table 2 considers the case of homogeneous hardcore in which Young’s modulus and the mass density of layer 1 are Ec = 380 GPa and ρc = 3 800 kg/m3 (P1 , alumina) at the top face and Em = 70 GPa and ρm = 2 707 kg/m3 Table 1

Comparisons of the natural fundamental frequency parameter of simply supported square power-law FGM plates of Type A with other theories (h/b = 0.1)

k

Theory

0

CPT FSDPT TSDPT SSDPT Elasticity[30] Present

1.873 1.824 1.824 1.824 – 1.824

59 42 45 52

0.5


1-0-1

2-1-2 59 42 45 52

45

1.873 1.824 1.824 1.824 – 1.824

1.471 1.441 1.444 1.444 1.446 1.444

57 68 24 36 14 24

1


1.262 1.240 1.243 1.243 1.244 1.243

5


10


2-1-1 59 42 45 52

45

1.873 1.824 1.824 1.824 – 1.824

1.512 1.481 1.484 1.484 1.486 1.484

42 59 08 18 08 08

38 31 20 35 70 20

1.320 1.297 1.300 1.300 1.301 1.300

0.958 0.942 0.945 0.946 0.944 0.945

44 56 98 30 76 98

0.943 0.925 0.928 0.928 0.927 0.928

21 08 39 75 27 39

1-1-1 59 42 45 52

45

1.873 1.824 1.824 1.824 – 1.824

1.542 1.510 1.512 1.512 1.508 1.506

64 35 53 58 41 35

23 29 11 23 81 11

1.371 1.346 1.348 1.348 1.335 1.333

0.991 0.978 0.981 0.982 0.981 0.981

90 70 84 07 03 84

0.952 0.939 0.942 0.943 0.940 0.942

44 62 97 32 78 97

2-2-1 59 42 45 52

45

1.873 1.824 1.824 1.824 – 1.824

1.549 1.516 1.519 1.519 1.521 1.519

03 95 22 27 31 21

50 37 88 94 11 29

1.375 1.350 1.353 1.353 1.355 1.353

1.087 1.071 1.074 1.074 1.029 1.030

97 56 32 45 42 43

1.051 1.035 1.038 1.045 0.989 0.991

85 80 62 58 29 95

1-2-1 59 42 45 52

45

1.873 1.824 1.824 1.824 – 1.824

1.583 1.550 1.551 1.552 1.549 1.547

74 01 99 02 26 10

1.607 1.572 1.574 1.574 1.576 1.574

22 74 51 50 68 51

21 72 33 39 23 32

1.432 1.405 1.407 1.407 1.397 1.395

47 55 89 92 63 57

1.464 1.437 1.439 1.439 1.441 1.439

97 22 34 31 37 33

1.055 1.041 1.044 1.044 1.045 1.044

65 83 66 81 32 66

1.161 1.144 1.147 1.147 1.109 1.108

95 67 31 41 83 81

1.188 1.171 1.173 1.173 1.175 1.173

67 59 97 99 67 97

1.005 0.992 0.995 0.995 0.995 0.995

24 56 51 19 23 50

1.118 1.102 1.105 1.041 1.061 1.060

83 61 33 54 04 90

1.136 1.120 1.123 1.134 1.124 1.123

14 67 14 60 66 14

45

938


Table 2

Comparison of fundamental frequency parameter of simply supported square power-law FGM sandwich plates with homogeneous hardcore

h/b

k 0 0.5

0.01

1 5 10 0 0.5

0.1

1 5 10 0 0.5

0.2

1 5 10

Theory

1-0-1

2-1-2

1-1-1

2-2-1

1-2-1

1-8-1

al.[30]

Li et Present Li et al.[30] Present Li et al.[30] Present Li et al.[30] Present Li et al.[30] Present

1.888 1.888 1.482 1.482 1.271 1.271 0.965 0.965 0.950 0.950

29 25 44 41 58 56 63 64 42 44

1.888 1.888 1.523 1.523 1.329 1.329 0.999 0.999 0.959 0.959

29 25 55 53 74 72 03 03 34 37

1.888 1.888 1.560 1.560 1.385 1.385 1.063 1.063 1.012 1.012

29 25 46 42 11 08 09 09 37 36

1.888 1.888 1.590 1.590 1.429 1.429 1.130 1.130 1.080 1.080

29 25 31 30 92 90 20 19 65 65

1.888 1.888 1.619 1.619 1.475 1.475 1.196 1.196 1.144 1.144

29 25 15 12 58 54 99 97 08 06

1.888 1.888 1.763 1.763 1.699 1.699 1.569 1.569 1.541 1.541

29 25 57 54 06 04 88 85 64 62

Li et al.[30] Present Li et al.[30] Present Li et al.[30] Present Li et al.[30] Present Li et al.[30] Present

1.826 1.824 1.446 1.444 1.244 1.243 0.944 0.945 0.927 0.928

82 45 14 23 70 19 76 98 27 38

1.826 1.824 1.486 1.484 1.301 1.300 0.981 0.981 0.940 0.942

82 45 08 08 81 10 03 84 78 96

1.826 1.824 1.521 1.519 1.355 1.353 1.045 1.044 0.995 0.995

82 45 31 21 23 32 32 65 23 50

1.826 1.824 1.549 1.547 1.397 1.395 1.109 1.108 1.061 1.060

82 45 26 10 63 56 83 81 04 90

1.826 1.824 1.576 1.574 1.441 1.439 1.175 1.173 1.124 1.123

82 45 68 50 37 32 67 96 66 13

1.826 1.824 1.711 1.709 1.651 1.648 1.529 1.527 1.503 1.501

82 45 30 01 13 92 93 92 33 38


1.677 1.670 1.353 1.347 1.174 1.169 0.890 0.894 0.868 0.871

11 10 58 43 85 76 86 62 33 78

1.677 1.670 1.390 1.384 1.229 1.223 0.933 0.935 0.892 0.899

11 10 53 10 15 40 62 94 28 18

1.677 1.670 1.421 1.415 1.277 1.271 0.997 0.995 0.949 0.950

11 10 78 08 70 34 98 45 84 33

1.677 1.670 1.445 1.438 1.314 1.307 1.056 1.052 1.009 1.008

11 10 35 43 34 53 07 28 49 48

1.677 1.670 1.469 1.462 1.353 1.346 1.119 1.113 1.072 1.067

11 10 40 51 41 71 00 18 90 54

1.677 1.670 1.581 1.574 1.531 1.524 1.428 1.421 1.405 1.399

11 10 86 76 42 45 45 97 68 32

(P2 , aluminum) at the bottom face. Table 3 considers the case of homogeneous softcore in which Young’s modulus and the mass density of Layer 1 are Em = 70 GPa and ρm = 2 707 kg/m3 (P1 , alumina) at the top face and Ec = 380 GPa and ρc = 3 800 kg/m3 (P2 , Aluminium) at the bottom face. Three thickness-side ratios h/b (0.01, 0.1, and 0.2) and five volume fraction indices k(0, 0.5, 1, 5, and 10) are considered. From the results presented in Tables 2 and 3, it can be seen that the fundamental frequencies of this study show a satisfied agreement with those obtained by Li et al.[30] . Table 4 gives the results of 1-8-1 power-law FGM plate of Type B. P1 is referred to as the properties of alumina and P2 the properties of aluminium. In this case, the FGM core is metal-rich at the top face and ceramic-rich at the bottom face. Three thickness-side ratios h/b (0.01, 0.1, and 0.2) and five volume fraction indices k (0.5, 1, 2, 5, and 10) are considered. Table 4 shows that the results of Li et al.[30] for FGM sandwich plates with the FGM core are in good agreement with the present RPT. From the results presented in Tables 2–4, it is shown that the natural fundamental frequencies decrease with the decrease of the material rigidity, which is due to the increase of k for Type A or the decrease of k for Type B and the variation of the layer thickness ratios. Moreover, the thin plates are slightly more sensitive than the thick plate to the material rigidity, i.e., k.


Comparison of fundamental frequency parameter of simply supported square power-law FGM sandwich plates with homogeneous softcore

Table 3 h/b

k 0 0.5 1

0.01

5 10 0 0.5 1

0.1

5 10 0 0.5 1

0.2

5 10

Table 4

939

Theory

1-0-1

2-1-2

1-1-1

2-2-1

1-2-1

1-8-1


0.960 0.960 1.662 1.662 1.820 1.820 1.920 1.920 1.910 1.910

22 20 81 83 31 34 90 89 64 61

0.960 0.960 1.622 1.622 1.791 1.791 1.943 1.943 1.946 1.947

22 20 91 94 63 74 13 32 87 01

0.960 0.960 1.581 1.581 1.753 1.753 1.936 1.936 1.950 1.950

22 20 71 73 79 91 23 58 44 80

0.960 0.960 1.522 1.522 1.681 1.681 1.862 1.862 1.880 1.880

22 20 77 79 84 94 07 39 42 76

0.960 0.960 1.506 1.506 1.674 1.674 1.885 1.885 1.911 1.911

22 20 58 57 90 94 30 58 62 98

0.960 0.960 1.265 1.265 1.383 1.383 1.570 1.570 1.604 1.604

22 20 57 55 31 30 35 34 57 56


0.928 0.927 1.573 1.574 1.722 1.725 1.841 1.841 1.840 1.838

97 76 52 97 27 68 98 99 20 57

0.928 0.927 1.525 1.528 1.674 1.683 1.826 1.841 1.839 1.851

97 76 88 95 37 79 11 61 87 96

0.928 0.927 1.484 1.486 1.630 1.639 1.789 1.817 1.808 1.836

97 76 59 66 53 66 56 30 13 65

0.928 0.927 1.434 1.436 1.570 1.578 1.727 1.753 1.747 1.775

97 76 19 15 37 74 26 20 79 27

0.928 0.927 1.416 1.416 1.557 1.561 1.726 1.748 1.748 1.775

97 76 62 26 88 02 70 64 11 84

0.928 0.927 1.205 1.204 1.308 1.307 1.466 1.466 1.494 1.494

97 76 53 77 25 66 47 00 81 39


0.852 0.849 1.378 1.382 1.508 1.517 1.658 1.658 1.672 1.667

86 27 94 25 96 15 68 29 78 89

0.852 0.849 1.320 1.327 1.433 1.455 1.580 1.617 1.609 1.639

86 27 61 72 25 15 11 77 09 13

0.852 0.849 1.280 1.285 1.382 1.403 1.502 1.566 1.526 1.592

86 27 53 21 42 11 84 07 71 71

0.852 0.849 1.245 1.249 1.342 1.361 1.460 1.520 1.483 1.547

86 27 33 99 03 64 09 42 06 63

0.852 0.849 1.225 1.224 1.321 1.328 1.426 1.474 1.441 1.501

86 27 80 81 29 28 65 40 01 43

0.852 0.849 1.070 1.068 1.144 1.143 1.252 1.251 1.270 1.270

86 27 16 52 51 53 10 56 65 17

Comparison of fundamental frequency parameter of simply supported square power-law FGM sandwich plates with the FGM core k

h/b

Theory

0.5

1

2

5

10

0.01

Li et al.[30] Present

1.339 31 1.339 27

1.386 69 1.386 65

1.444 91 1.444 87

1.531 43 1.531 39

1.591 05 1.591 03

0.1


1.297 51 1.294 59

1.348 47 1.345 33

1.408 28 1.405 14

1.493 09 1.490 44

1.549 80 1.547 54

0.2


1.195 80 1.186 82

1.253 38 1.243 52

1.315 69 1.305 76

1.395 67 1.387 36

1.445 40 1.438 37

Figure 3 depicts the fundamental frequency parameters versus the thickness-side ratios of simply supported power-law FGM sandwich plates with the homogeneous hardcore. Figure 4 depicts the curves of the power-law FGM sandwich plates with the homogeneous softcore. The results are the maximum for the ceramic plates and the minimum for the metal plates. It is seen that the results increase smoothly as the amount of ceramic in the sandwich plate increases. It is also shown that the effect of k on the 1-0-1 sandwich plate without the homogeneous core layer is greater than that of the 1-8-1 sandwich with the homogeneous hardcore, and the effect of k on the sandwich with the homogeneous hardcore is greater than that with the homogeneous

940


softcore.

Fig. 3

Fundamental frequencies for power-law FGM sandwich plates with homogeneous hardcore

Fig. 4

Fundamental frequencies for power-law FGM sandwich plates with homogeneous softcore

4

Concluding remarks

A four-variable RPT is developed for the vibration analysis of the rectangular FG sandwich. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the FG sandwich plate. Hence it is unnecessary to use shear correction factors. The power-law FGM sandwich plates with the FGM facesheet and the homogeneous core and the sandwich plates with the homogeneous facesheet and the FGM core are considered. The governing equations have strong similarity with the CPT in many aspects. All comparison studies demonstrate that the present solution is highly efficient for the exact analysis of the vibration of FG rectangular sandwich plates. In conclusion, it can be said that the proposed theory RPT is accurate and simple in solving the free vibration behavior of FG sandwich plates. However, it can be noted that the improvement of the present theory will be necessary, especially when it is applied to a laminated structure to satisfy interlayer transverse shear stress continuity. The extension of the present theory is also envisaged for general boundary conditions.


941

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