Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories

Composites Part B 83 (2015) 203e215

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Composites Part B journal homepage: www.elsevier.com/locate/compositesb

Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories S.S. Akavci a, *, A.H. Tanrikulu b, 1 a b

Department of Architecture, University of Cukurova, 01330, Balcali, Adana, Turkey Department of Civil Engineering, University of Cukurova, 01330, Balcali, Adana, Turkey

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 January 2015 Received in revised form 5 June 2015 Accepted 6 August 2015 Available online 21 August 2015

In this study, two dimensional (2D) and quasi three-dimensional (quasi-3D) shear deformation theories are presented for static and free vibration analysis of single-layer functionally graded (FG) plates using a new hyperbolic shape function. The material of the plate is inhomogeneous and the material properties assumed to vary continuously in the thickness direction by three different distributions; power-law, exponential and MorieTanaka model, in terms of the volume fractions of the constituents. The fundamental governing equations which take into account the effects of both transverse shear and normal stresses are derived through the Hamilton's principle. The closed form solutions are obtained by using Navier technique and then fundamental frequencies are found by solving the results of eigenvalue problems. In-plane stress components have been obtained by the constitutive equations of composite plates. The transverse stress components have been obtained by integrating the three-dimensional stress equilibrium equations in the thickness direction of the plate. The accuracy of the present method is demonstrated by comparisons with the different 2D, 3D and quasi-3D solutions available in the literature. © 2015 Elsevier Ltd. All rights reserved.

Keywords: A. Layered structures B. Mechanical properties C. Analytical modeling D. Numerical analysis

1. Introduction Functionally graded materials (FGMs) are a type of advanced composite materials whose properties vary gradually and continuously from one surface to another. The mechanical properties of FGM vary along the thickness direction in the material depending on a function. Due to this feature, the FGMs have some advantages such as eliminating the material discontinuity and avoiding the delamination failure, reducing the stress levels and deflections. Combination of these characteristics attracts application of FGMs in many engineering fields from biomedical to civil engineering. In recent years, FGM applications have received considerable increase. The increase in FGM applications requires accurate models to predict their response. The static and dynamic behavior of FGM plates have been analyzed and reported by many researchers in recent ten years. Kashtalyan [1] developed a threedimensional elasticity solution for a simply supported FG plate

* Corresponding author. Tel.: þ90 322 3387230; fax: þ90 322 3386126. E-mail addresses: [email protected] (S.S. Akavci), [email protected] (A.H. Tanrikulu). 1 Tel.: þ90 322 3386226; fax: þ90 322 3386126. http://dx.doi.org/10.1016/j.compositesb.2015.08.043 1359-8368/© 2015 Elsevier Ltd. All rights reserved.

subjected to transverse loading. Tornabene et al. [2] analyzed the dynamic behavior of FG conical, cylindrical shells and annular plates. They used the first-order shear deformation theory and the linear elasticity theory to analyze moderately thick structural elements. Matsunaga [3] presented a two-dimensional higher-order theory for the evaluation of displacement and stresses in FGM plates subjected to thermal and mechanical loadings. The theory can take into account the effects of both transverse shear and normal stresses. Zhao et al. [4] presented a free vibration analysis of metal and ceramic FG plates that uses the element-free kp-Ritz method. The first-order shear deformation plate theory is employed to account for the transverse shear strain and rotary inertia, and mesh-free kernel particle functions are used to approximate the two-dimensional displacement fields. Vaghefi et al. [5] developed a version of meshless local PetroveGalerkin method to obtain 3D static solutions for thick FG plates. Orakdogen et al. [6] studied the coupling effect of extension and bending in FG plate subjected to transverse loading for Kirchhoff-Love plate theory equations. Tamijani and Kapania [7] developed a element free Galerkin method for the free vibration of a FG plate with curvilinear stiffeners. The governing equations for the plate and stiffeners are derived by using the first order shear deformation theory. Gunes

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et al. [8] carried out an experimental work about the low-velocity impact behavior of FG clamped circular plates. Zhang et al. [9] presented a 3D elasticity solution for static bending of thick FG plates using a hybrid semi-analytical approach-the state-space based differential quadrature method. Filippi et al. [10] used the 1D Carrera Unified Formulation to perform static analyses of FG structures. Swaminathan et al. [11] presented a comprehensive review of the various methods employed to study the static, dynamic and stability behavior of FGM plates. Barretta et al. [12] analyzed the torsion of linearly elastic isotropic beams, with both cross-sectional and axial inhomogeneities. New closed-form solutions are found in the present paper, by detecting axial distributions of longitudinal and shear moduli inducing an axially uniform warping field. Sofiyev and Kuruoglu [13] presented a theoretical approach to solve vibration problems of FG truncated conical shells under mixed boundary conditions. Zafarmand and Kadkhodayan [14] investigated the three dimensional static and dynamic behavior of a thick sector plate made of two-directional FGMs. Zhu and Liew [15] presented free vibration analyses of metal and ceramic FG plates with the local Kriging meshless method. It is obvious that non-negligible shear deformations occur at the thick and moderately thick plates and the classical plate theory shows inaccurate results. So, transverse shear deformations have to be taken into account in the analysis. There are numerous plate theories that include transverse shear strains for the advanced composites like FGMs. Zenkour [16] analyzed the static response of simply supported FG rectangular plates subjected to transverse uniform load. Bodaghi and Saidi [17] presented an analytical approach which converts the coupled governing stability equations into two uncoupled partial differential equations in terms of transverse displacement for buckling analysis of thick FG rectangular plates. Benachour et al. [18] presented a four variable refined plate theory for free vibration analysis of plates made of FGMs with an arbitrary gradient. Hosseini-Hashemi et al. [19] presented a new exact closed-form procedure to solve free vibration analysis of FG rectangular thick plates based on the Reddy's third-order shear deformation plate theory while the plate has two opposite edges simply supported. Thai and Kim [20] developed a new shear deformation theory for bending and free vibration analysis of FG plates. Tran et al. [21] presented a novel and effective formulation based on isogeometric approach and higher-order deformation plate theory to study the behavior of FGM plates. Mechab et al. [22] presented the analytical solutions of static and dynamic analysis of FG plates using a four-variable refined plate theory. Mantari and Soares [23] presented static response of FG plates by using a recently developed higher order shear deformation theory. The thickness stretching effect is ignored in the most of shear deformation theories by assuming the transverse displacement as constant. This assumption is inaccurate especially for thick FGM plates. To overcome this problem, some quasi-3D theories presented in the literature. Zenkour [24] considered the bending problem of transverse load acting on an isotropic inhomogeneous rectangular plate using both 2D trigonometric and 3D elasticity solutions. Matsunaga [25] analyzed the natural frequencies and buckling stresses of plates made of FGMs by taking into account the effects of transverse shear and normal deformations and rotatory inertia. Carrera et al. [26] evaluated the effect of thickness stretching in plate/shell structures made by materials which are FGM in the thickness directions. Neves et al. [27] presented a new application for Carrera's unified Formulation (CUF) to analyze FG plates. Neves et al. [28] presented an original hyperbolic sine shear deformation theory which account for through the thickness deformations for the bending and free vibration analysis of FG plates. Zenkour [29] derived a refined trigonometric higher-order plate theory which the effects of transverse shear strains as well as the

transverse normal strain are taken into account for bending analysis of FGM plates. Jha et al. [30] presented a free vibration response of FG elastic, rectangular, and simply (diaphragm) supported plates based on higher order shear/shear-normal deformations theories. Sheikholeslami and Saidi [31] studied the free vibration of simply supported FG rectangular plates resting on two-parameter elastic foundation using the higher-order shear and normal deformable plate theory of Batra and Vidoli by an analytical approach. Thai and Kim [32] developed a quasi-3D sinusoidal shear deformation theory for FG plates. Mantari and Soares [33] presented the analytical solutions to the bending analysis of FGM plates by using a new trigonometric higher order and hybrid quasi-3D shear deformation theories. Hebali et al. [34] presented a new quasi-3D hyperbolic shear deformation theory for bending and free vibration analysis of FG plates. Thai and Choi [35] improved a refined pate theory to account for the effect of thickness stretching in FG plates. Mantari and Soares [36] presented a quasi-3D trigonometric shear deformation theory for the bending analysis of FG plates. Belabed et al. [37] presented an efficient and simple higher order shear and normal deformation theory for FGM plates. Alijani and Amabili [38] investigated the nonlinear forced vibrations of moderately thick FG rectangular plates by considering higher-order shear deformation theories that take into account the thickness deformation effect. In this study, a new quasi-3D hyperbolic shear deformation theory is developed for static and free vibration analysis of FG plates. The proposed theory gives good accuracy and accounts for a parabolic transverse shear deformation shape function and satisfies shear stress free boundary conditions of top and bottom surfaces of the plate without using shear correction factors. Besides, the theory accounts for the thickness stretching effect by using the same hyperbolic function. Governing equations are derived from the Hamilton's principle. Navier solution is used to obtain the closed form solutions for simply supported functionally graded plates. The in-plane stresses are calculated from the linear constitutive relations and the transverse shear and normal stresses are obtained by integrating the three dimensional stress equilibrium equations of elastic media by satisfying the stress boundary conditions on the top and bottom surfaces of a plate. Numerical examples are presented to illustrate the accuracy and efficiency of the present theory. 2. Fundamental formulations 2.1. Homogenization models for continuous gradation Consider a single-layer rectangular plate, having uniform thickness h, length a, width b and made of a functionally graded material (Fig. 1). In this study, the compositions and volume fractions of the constituents in FGM are considered to vary gradually through the thickness according to: a. The power-law distribution, b. The exponential distribution, c. The MorieTanaka homogenization model. Since the effects of Poisson's ratio n on the response of FG plates are very small, it is assumed to be constant for all gradation models. a. The power-law (P-FGM) distribution The volume fraction of the P-FGM plate is assumed to vary continuously through the thickness of the plate in according to the power law distribution (Bao and Wang [39]) as follows:


PðzÞ ¼ Pm þ ðPm  Pc Þ

 1 z k þ 2 h

(1)

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

z

In literature, FGM plates have been analyzed mostly by neglecting the thickness stretching effect. The effect of thickness stretching in FGM plates has been investigated by Zenkour [24], Matsunaga [25] and Carrera et al. [26]. Then some researchers used the same displacement field in Equation (4) with different types of the shape functions. The shape functions and their derivatives used by several other researchers can be summarized in Table 1. In this study, the shape function in Equation (4) is expressed by a hyperbolic function and assures an accurate distribution of shear deformation through the plate thickness (Fig. 2) and allows to transverse shear stresses vary as parabolic across the thickness as satisfying shear stress free surface conditions without using shear correction factors. This hyperbolic shear deformation theory accounts for not only shear deformations but also the thickness stretching effect. In addition, this formula is suitable for different FGM plates and easy to implement. The numerical examples show that the present theories show good agreement with that of the results of other 2D and quasi-3D shear deformation theories. Also, the results predicted by the proposed theories are in an excellent agreement with 3D elasticity solutions even for the case of very thick plates. The chosen hyperbolic shape function in this study is

a b

x

h y z

Ceramic surface

h

205

x Metallic surface

Fig. 1. Geometry and coordinates of the considered single-layer FGM plate.

b. The exponential (E-FGM) distribution The volume fraction of the E-FGM plate is assumed to vary continuously through the thickness of the plate in according to the exponential distribution (Delale and Erdogan [40]) as follows:

PðzÞ ¼ Ae

pðzþ2h Þ


 1 Pc ; A ¼ Pm ; p ¼ ln h Pm

(2)

c. The Mori Tanaka Homogenization Model For MorieTanaka scheme, the volume fraction of the FGM plate is given as (Mori and Tanaka [41], Benveniste [42]):




PðzÞ ¼ Pm þ ðPc  Pm Þ 1 þ Vm Vc ¼ ð0:5 þ z=hÞ

Vc

 Pc  1 Pm

; Vm ¼ 1  Vc ; 1þy 33y

(3)

  df ðzÞ f ðzÞ ¼ 3:7z 1:27Sech0:65 ðz=hÞ  1 and f 0 ðzÞ ¼ dz

Based on the assumptions in Eq. (4), within the application of the linear, small-strain elasticity theory, the general strainedisplacement relations are defined as

(

gyz

xy

) ¼

gxz

v2 w0

kqx ¼

2.2. Kinematics

kqyz ¼ qy þ

9 8 9 8 < uðx; y; zÞ = < u0 ðx; yÞ  zw0 ðx; yÞ;x þ f ðzÞqx ðx; yÞ = vðx; y; zÞ ¼ v ðx; yÞ  zw0 ðx; yÞ;y þ f ðzÞqy ðx; yÞ ; : ; : 0 wðx; y; zÞ w0 ðx; yÞ þ f 0 ðzÞqz ðx; yÞ

(4)

where, u, v, w are displacements in the x, y, z directions, u0, v0 and w0 are mid-plane displacements, qx, qy and qz are rotations of normals to the midplane about y-axis, x-axis, and z-axis, respectively, and a comma followed by a variable denotes differentiation with respect to that variable. f(z) is the hyperbolic shape function which represents the distribution of the transverse shear displacements along the thickness. Note that f0 (z) ¼ 0 for 2D analysis.

kx ¼

df ðzÞ dz

9 8 q 9 8 kx > kx > > > > > > > = = < < þ f ðzÞ kqy  z ky > > > > > > > > > ; ; > : q > : ; kxy kxy

(

kqyz

)

; εz ¼

kqxz

d2 f ðzÞ dz2

qz

vu0 0 vv0 0 vu vv ;ε ¼ ;g ¼ 0 þ 0 ; vx y vy xy vy vx

where P denotes the effective material property like Young's modulus E, Pm and Pc denotes the property of the top and the bottom faces of the plate, respectively, k is the power law index and p is the volume fraction exponent.

On the basis of the thick plate theory and including the effect of transverse normal stress (thickness stretching effect), the assumed displacement field of the plate can be described as

9 > > > =

8 9 8 0 > εx > > > εx > > < = > < εy ε0 ¼ > > > y > > : ; > > : 0 gxy g

ε0x ¼

k

(5)

vx

2

; ky ¼

v2 w0 vy

2

; kxy ¼ 2

(6)

v2 w0 ; vxvy

vqx q vqy q vqx vqy ;k ¼ ;k ¼ þ ; vx y vy xy vy vx vqz q vqz ; k ¼ qx þ vy xz vx

The linear constitutive relations of a composite plate according to the three dimensional elasticity can be expressed as

8 sxx > > > > > < syy szz > txy > > > > : tyz txz

9 > > > > > =

2

Q11 6 Q12 6 6 Q12 ¼6 6 > > 6 0 > > 4 0 > ; 0

Q12 Q11 Q12 0 0 0

Q12 Q12 Q11 0 0 0

0 0 0 Q44 0 0

0 0 0 0 Q55 0

3 0 8 εxx > > > εyy 0 7 > 7> < 0 7 7 εzz gxy 0 7 > 7> > g > 5 0 > : yz gxz Q66

9 > > > > > = > > > > > ;

(7)

in which (sxx, syy, szz,txy, tyz, txz) and (εx, εy, εz, gxy, gyz, gxz) are the stress and strain components, respectively. The elastic constants (Qij's) are depends on the normal strain εzz. If the εzz s 0 then Qij's are;

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S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

Table 1 The forms of shape function f(z) and corresponding transverse displacement w for different shear deformation theories. Theory

f(z)

f0 (z)

w

Zenkour [24]

h pz p sin h

cos pz h

w0 þ f (z)qz

Carrera et al. [26]

z,

e

w0 þ zw1 þ z2w2 þ / þ znwn

Neves et al. [27]

sin pz h sinh pz h

i

Neves et al. [28] Zenkour [29]

i ¼ 1,N

e




h sinh hz  34 hz 2 cosh

2

1 2

cosh hz  4hz2

h pz p sin h h sinh pz z p h cosh p21

Hebali et al. [34] Thai and Choi [35]

3

ze

2

1 f 0 ðzÞq w0 þ 12 z 0

cos pz h

w0 þ f (z)qz

cosh pz 1 h cosh p21

wb þ ws þ (1f'(z))qz wb þ ws þ (1f'(z))qz

14 þ 5z h2  2 4z 1  h2

 2

Karama et al. [47]

w0 þ zw1 þ z2w2


 cosh 12

2

5z 4z þ 3h 2   2 4 z 1  3 hz

Reddy [46]

w0 þ zw1 þ z2w2

e 3

Thai and Kim [32]

0

 2 2

z h

e

Q11 ¼ EðzÞð1  yÞ=ð1  2yÞð1 þ yÞ Q12 ¼ EðzÞy=ð1  2yÞð1 þ yÞ Q44 ¼ Q55 ¼ Q66 ¼ GðzÞ GðzÞ ¼ EðzÞ=2ð1 þ yÞ

Zt2 d

(8)

z h

w0 2 2z 2

 4e hh2

w0 z2

  V þ Vp  T dt ¼ 0

(10)

t1

If the εzz ¼ 0 then Qij's are;

. Q11 ¼ EðzÞ ð1  y2 Þ . Q12 ¼ EðzÞy ð1  y2 Þ Q44 ¼ Q55 ¼ Q66 ¼ GðzÞ GðzÞ ¼ EðzÞ=2ð1 þ yÞ

where; V is the strain energy of the plate and Vp is the potential energy of applied distributed transverse load and T is the kinetic energy of FG plate. The governing equations of equilibrium can be derived from Equation (10) by integrating the displacement gradients by parts and setting the coefficients du, dv, dw, dqx, dqy, and dqz zero separately:

(9)

€ 0  I2 w € 0;x þ I4 € Nx;x þ Nxy;y ¼ I1 u qx € 0;y þ I4 € Nxy;x þ Ny;y ¼ I1 € qy v0  I2 w       € 0;yy þ I5 € € 0;x þ € € 0;xx þ w € 0 þ I2 u Mx;xx þ 2Mxy;xy þ My;yy þ q ¼ I1 w v0;y  I3 w qy;y þ I7 € qz qx;x þ € € 0  I5 w € 0;x þ I6 € qx Px;x þ Pxy;y  Rx ¼ I4 u € 0;y þ I6 € Pxy;x þ Py;y  Ry ¼ I4 € qy v0  I5 w 0 € 0 þ I8 € qz Rx;x þ Ry;y  Sz þ f q ¼ I7 w

where the stress, moment resultants, stiffness components and inertias are defined in Appendix. The in-plane normal and shear stresses (sx, sy and txy) can be obtained accurately by the constitutive relations (7) for composite plates. But if the transverse normal and shear stresses (sz, tyz and txz) calculated from the constitutive relations (7), they may not

2.3. Equations of equilibrium and stress components The Hamilton principle is used herein to derive the equations of motion appropriate to the displacement field and the constitutive equations. The principle can be stated in analytical form as:

z/h

0.5 0.4 0.3 0.2 0

Present Study Zenkour [24]

-0.2

-0.7

-0.5

-0.3

z/h

0.1 -0.1 -0.3

Reddy [46]

-0.4

Karama [47]

-0.5 -0.1

0.1

f(z)

0.3

0.5

0.7

(11)

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

0.2

0.4

0.6

0.8

f '(z)

Fig. 2. The Shape function f(z) and it's differentiation f0 (z) for different shear deformation theories.

1

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

satisfy the boundary conditions at the top and bottom surfaces of the plate. So these stresses are obtained by integrating the equilibrium equations of three-dimensional elasticity with respect to thickness coordinate as:

 Zz
vsx vtxy þ dz þ C1 ðx; yÞ vx vy

txz ¼ 

(12)

h=2

Zz
tyz ¼ 

 vtxy vsy þ dz þ C2 ðx; yÞ vx vy

Zz sz ¼ 

Zz "

B @

h=2

h=2

8 < 16q0 m; n ¼ 1; 3; 5; … For uniformly distributed load; qmn ¼ mnp2 : 0 m; n ¼ 2; 4; 6; … For sinusoidal distributed load qmn ¼ q0 in which q0 is the intensity of the load. Substituting the stress and moment resultants defined in Appendix and the Equation (17) into equations of motion (11) we get below closed-form solutions of bending and free vibration problems of FGM plate;

(13) 

h=2

0

1

 ½K  u2 ½M fDg ¼ fPg

# v2 txy v2 sy v2 sx C þ þ 2 dzAdz þ C3 ðx; yÞz vxvy vx2 vy2

þ C4 ðx; yÞ (14) where Ci's (i ¼ 1,4) are constants and determined by the following boundary conditions at the top and bottom surfaces of the plate:

2 6 6 6 6 ½K ¼ 6 6 6 6 4

txz jz¼±h=2 ¼ 0; tyz z¼±h=2 ¼ 0; sz jz¼h=2 ¼ qðx; yÞ; sz jz¼h=2 ¼ 0 (15) fPg ¼ 3. Analytical solution for rectangular FGM plates For a simply supported rectangular FG plate with length a and width b, the kinematic boundary conditions are given below:

Nx ¼ v ¼ w ¼ Mx ¼ Px ¼ Ry ¼ Sz ¼ qy ¼ qz ¼ 0 at x ¼ 0; a Ny ¼ u ¼ w ¼ My ¼ Py ¼ Rx ¼ Sz ¼ qx ¼ qz ¼ 0 at y ¼ 0; b (16) For the analytical solution of the partial differential equation (11), the Navier method, based on double Fourier series, is used for the simply supported boundary conditions (16). Using Navier's procedure, the solution of the displacement variables satisfying the above boundary conditions can be expressed in the following Fourier series:

∞ P ∞ P

uðx;yÞ ¼

Amn cos

m¼1 n¼1 ∞ P ∞ P

vðx;yÞ ¼

Bmn sin

m¼1 n¼1

wðx;yÞ ¼

∞ P ∞ P

qðx; yÞ ¼

∞ X

m¼1 n¼1

a12

a13

a14

a15

a22

a23 a33

a24 a34

a25 a35

a44

a45 a55

SYM

8 > > > > > > > > < > > > > > > > > : 2

6 6 6 6 6 ½M ¼ 6 6 6 6 4

0 0 q

9 > > > > > > > > =

8 Amn > > > > > a26 7 Bmn > 7 > > 7 Txmn a46 7 > 7 > > 7 > > a56 5 > Tymn > : Tzmn a66 a16

3

> > > > > > 0 > > ; 0 f ðz ¼ h=2Þq

I1

9 > > > > > > > > = ; > > > > > > > > ;

0

0

aI2

I4

0

I1

bI2 I1 þ I3 ða2 þ b2 Þ

0 I5 a

I4 I5 b

I6

0 I6

SYM

0

3

7 07 7 I7 7 7 7 07 7 07 5 I8 (20)

in which

∞ X ∞ X mpx npy iut mpx npy iut cos e ; qy ¼ cos e Tymn sin a b a b m¼1 n¼1

(17)

∞ X ∞ X mpx npy iut mpx npy iut sin e ; qz ¼ sin e Tzmn sin a b a b m¼1 n¼1

where Amn, Bmn, Cmn, Txmn, Tymn, Tzmn are arbitrary parameters to be determined and u is the natural frequency. The transverse distributed load q(x,y) is also expanded double Fourier series as; ∞ X

a11

(19)

∞ X ∞ X mpx npy iut mpx npy iut sin e ; qx ¼ sin e Txmn cos a b a b m¼1 n¼1

Cmn sin

m¼1 n¼1

207

qmn sin

mpx npy sin a b

(18)

The coefficients qmn are given below for some general loadings;

Table 2 Material properties used in the functionally graded plates. Material

Alimunium (Al) Alumina (Al2O3) Zirconia (ZrO2)

Properties E (GPa)

n

r (kg/m3)

70 380 200

0.3 0.3 0.3

2702 3800 5700

208

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

Table 3 The non-dimensional displacement and stress components of an Al/Al2O3 FGM square plate subjected to uniformly distributed load (a/h ¼ 10). k

Theory

εz

wð0Þ

0

Zenkour [16] Present Study Present Study Zenkour [16] Present Study Present Study Zenkour [16] Present Study Present Study Zenkour [16] Present Study Present Study Zenkour [16] Present Study Present Study

¼0 ¼0 s0 ¼0 ¼0 s0 ¼0 ¼0 s0 ¼0 ¼0 s0 ¼0 ¼0 s0

0.4665 0.4665 0.4635 0.9287 0.9288 0.8977 1.1940 1.1940 1.1376 1.3890 1.3888 1.3259 1.5876 1.5875 1.5453

1

2

4

10


 sx

h 2

2.8932 2.8909 2.9981 4.4745 4.4707 4.6110 5.2296 5.2248 5.3825 5.8915 5.8855 6.0382 7.3689 7.3617 7.5123


 sy

h 3

1.9103 1.9103 1.8925 2.1692 2.1693 2.0822 2.0338 2.0342 1.9257 1.7197 1.7205 1.6062 1.2820 1.2828 1.2016

txz ð0Þ 0.5114 0.4988 0.4782 0.5114 0.4988 0.4782 0.4700 0.4581 0.4524 0.4204 0.4090 0.4358 0.4552 0.4436 0.4332


 tyz

h 6


txy

  h3

0.4429 0.4363 0.4315 0.5446 0.5364 0.5119 0.5734 0.5643 0.5081 0.5346 0.5253 0.4804 0.4227 0.4159 0.4561

1.2850 1.2857 1.2578 1.1143 1.1141 1.0211 0.9907 0.9909 0.8921 1.0298 1.0305 0.9274 1.0694 1.0705 0.9860

Table 4 Non-dimensional displacement and stress of an Al/Al2O3 FGM square plate subjected to sinusoidal load. k

Theory

1

4

10

Carrera et al. [26] Neves et al. [27] Neves et al. [28] Zenkour [29] Hebali et al. [34] Present Study Present Study Carrera et al. [26] Neves et al. [27] Neves et al. [28] Zenkour [29] Hebali et al. [34] Present Study Present Study Carrera et al. [26] Neves et al. [27] Neves et al. [28] Zenkour [29] Hebali et al. [34] Present Study Present Study

εz

s0 s0 s0 s0 s0 ¼0 s0 s0 s0 s0 s0 s0 ¼0 s0 s0 s0 s0 s0 s0 ¼0 s0


 sx

wð0Þ

h 3

a/h ¼ 4

a/h ¼ 10

a/h ¼ 100

a/h ¼ 4

a/h ¼ 10

a/h ¼ 100

0.6221 0.5925 0.5910 0.5944 0.5952 0.5806 0.5754 0.4877 0.4404 0.4340 0.4321 0.4507 0.4431 0.4247 0.3965 0.3227 0.3108 0.3154 0.3325 0.3242 0.3095

1.5064 1.4945 1.4917 1.4962 1.4954 1.4895 1.4322 1.1971 1.1783 1.1593 1.1410 1.1779 1.1787 1.1017 0.8965 1.1783 0.8467 0.8530 0.8889 0.8778 0.8229

14.969 14.969 14.944 14.552 14.963 14.967 14.306 11.923 11.932 11.738 11.388 11.871 11.920 11.088 8.9077 11.932 8.6013 8.5853 8.9977 8.9059 8.3185

0.7171 0.6997 0.7020 0.6828 0.6910 0.7282 0.6908 1.1585 1.1178 1.1095 1.1001 1.0964 1.1613 1.0983 1.3745 1.3490 1.3327 1.3391 1.3333 1.3917 1.3352

0.5875 0.5845 0.5868 0.5592 0.5686 0.5889 0.5691 0.8821 0.8750 0.8698 0.8404 0.8413 0.8818 0.8417 1.0072 0.8750 0.9886 0.9806 0.9791 1.0089 0.9818

0.5625 0.5624 0.5648 0.5624 0.5452 0.5625 0.5457 0.8286 0.8286 0.8241 0.7933 0.7926 0.8287 0.7925 0.9361 0.8286 0.9228 0.9140 0.9114 0.9362 0.9141

4. Numerical results

a11 a12 a13 a14 a15 a16 a22 a23 a24 a25 a26 a33 a34 a35 a36 a44 a45 a46 a55 a56 a66

¼ A11 a2  A34 b2 ¼ abðA12 þ A34 Þ ¼ B11 a3 þ B12 ab2 þ 2B34 ab2 ¼ C11 a2  C34 b2 ¼ abðC12 þ C34 Þ ¼ D13 a ¼ A34 a2  A22 b2 ¼ B21 a2 b þ 2B34 a2 b þ B22 b3 ¼ abðC21 þ C34 Þ ¼ C34 a2  C22 b2 ¼ D23 b ¼ E11 a4  E12 a2 b2  E21 a2 b2  4E34 a2 b2  E22 b4 ¼ F11 a3 þ F21 ab2 þ 2F34 ab2 ¼ F12 a2 b þ 2F34 a2 b þ F22 b3 ¼ G13 a2  G23 b2 ¼ L11  H11 a2  H34 b2 ¼ abðH12 þ H34 Þ ¼ K13 a  L11 a ¼ L22  H34 a2  H22 b2 ¼ K23 b  L22 b ¼ P13  L11 a2  L22 b2

np and a ¼ mp a ;b ¼ b .

In order to verify the accuracy of the present theory in predicting the static response and free vibration of simply supported advanced composite plates, different examples are solved and compared with the results of various 3D, quasi-3D and 2D shear deformation theories. 4.1. Static analysis

(21)

4.1.1. Functionally graded plates In this section, the calculated stress and displacements of FGM plates which are graded from the bottom to the top surface according to Equation (1) are given as compared with the results of different shear deformation theories. The material properties of FGM plates are listed in Table 2. In Tables 3 and 4, the non-dimensional displacement and stresses of an Al/Al2O3 FGM square plate subjected to uniformly and sinusoidal distributed loads are presented for different values of the power-law index. Table 3 shows the calculated results of nondimensional displacement and stress components of the square FGM plate under uniform load as compared with the published results of a generalized 2D shear deformation theory by Zenkour [16],

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215 0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.1 Present Study, k=4

z/h

z/h

0.2

Present Study, k=0

0.1 0.0

Present Study, k=10

-0.1

0.0 -0.1

Matsunaga[3], k=0

-0.2

-0.2

-0.3

Matsunaga[3], k=4

-0.3

-0.4

Matsunaga [3], k=10

-0.4

-0.5

-0.5 -30

-20

-10

0

10

20

30

40

50

60

0

0.25

0.5

0.75

1

σz

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

z/h

z/h

σx

0.0

0.0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4 -0.5

-0.5 -30

-20

-10

0

10

0

20

1

1

2

2

3

3

τxz

τxy 0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

z/h

z/h

209

0

0

-0.1

-0.1

-0.2

-0.2 -0.3

-0.3

-0.4

-0.4

-0.5

-0.5 -150

-50

50

150

250

0

500

1000

1500

w

u

Fig. 3. The distributions of the non-dimensional displacement and stresses of FGM plate (a/b ¼ 1, a/h ¼ 10).

where εz ¼ 0. It can be seen from the table that the present 2D theory results are in excellent agreement with the 2D theory results of Zenkour [16]. In addition, the present quasi-3D theory yields more accurate results than those obtained by the other two theories. This table also shows that, the transverse displacement w and in-plane stresses sx and sy increase and the shear stresses txz and txy decrease with the increasing value of power law index k. Table 4 presents the non-dimensional in-plane stresses sx and non-

dimensional transverse displacements w of a square plate for different a/h ratios. The present results are compared with the different quasi-3D higher order shear deformation theories of Carrera et al. [26], Neves et al. [27,28], Zenkour [29], Hebali et al. [34] that include both transverse shear and normal deformations. The present quasi-3D results agree very well with those given by other theories. The non-dimensional displacement and stress components for Tables 3 and 4:

  


a b h a b h a b ; ; z ; ; ; z ; ; ; z ; w s ðzÞ ¼ s s ðzÞ ¼ s x x y y 2 2 aq0 2 2 aq0 2 2 a4 q0 
 a h h b h txy ðzÞ ¼ txy ð0; 0; zÞ; txz ðzÞ ¼ txz 0; ; z ; tyz ðzÞ ¼ tyz ; 0; z aq0 aq0 2 aq0 2 wðzÞ ¼

10h3 Ec

210

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215


 Table 5 3 E0 Non-dimensional deflection wð0Þ ¼ 10h w 2a ; 2b ; 0 of EGM plates subjected to sinusoidal distributed load (a/h ¼ 2). a4 q 0

b/a

Theory

εz

1

Zenkour [24] Zenkour [24] Mantari and Soares Mantari and Soares Thai and Choi [35] Present Study Present Study Zenkour [24] Zenkour [24] Mantari and Soares Mantari and Soares Thai and Choi [35] Present Study Present Study Zenkour [24] Zenkour [24] Mantari and Soares Mantari and Soares Thai and Choi [35] Present Study Present Study

s0 ¼0 s0 ¼0 s0 ¼0 s0 s0 ¼0 s0 ¼0 s0 ¼0 s0 s0 ¼0 s0 ¼0 s0 ¼0 s0

2

3

[33] [33]

[33] [33]

[33] [33]

p 0.1

0.3

0.5

0.7

1.0

1.5

0.5769 0.5730 0.5778 0.6362 0.5777 0.6351 0.5750 1.1944 1.1879 1.1940 1.2776 1.1939 1.2763 1.1907 1.4429 1.4354 1.4421 1.5340 1.4419 1.5327 1.4386

0.5247 0.5180 0.5224 0.5751 0.5222 0.5741 0.5198 1.0859 1.0739 1.0794 1.1553 1.0793 1.1541 1.0765 1.3116 1.2977 1.3037 1.3873 1.3035 1.3861 1.3005

0.4766 0.4678 0.4717 0.5194 0.4716 0.5185 0.4694 0.9864 0.9700 0.9750 1.0441 0.9749 1.0431 0.9723 1.9112 1.1722 1.1776 1.2540 1.1774 1.2530 1.1748

0.4324 0.4221 0.4256 0.4687 0.4255 0.4679 0.4236 0.8952 0.8754 0.8799 0.9430 0.8798 0.9422 0.8775 1.0811 1.0579 1.0627 1.1329 1.0626 1.1320 1.0602

0.3726 0.3611 0.3648 0.4017 0.3640 0.4004 0.3624 0.7726 0.7493 0.7537 0.8092 0.7530 0.8079 0.7511 0.9333 0.9056 0.9104 0.9725 0.9096 0.9712 0.9076

0.2890 0.2771 0.2793 0.3079 0.2793 0.3075 0.2780 0.6017 0.5757 0.5786 0.6237 0.5785 0.6234 0.5771 0.7275 0.6961 0.6992 0.7506 0.6991 0.7503 0.6976



 Table 6 2 Non-dimensional stress sx 2h ¼ ah2 q sx 2a; 2b; 2h of EGM plates subjected to sinusoidal distributed load (a/h ¼ 10). 0

b/a

Theory

εz

1

Thai and Kim [32] Mantari and Soares Mantari and Soares Present Study Present Study Thai and Kim [32] Mantari and Soares Mantari and Soares Present Study Present Study Thai and Kim [32] Mantari and Soares Mantari and Soares Present Study Present Study

s0 s0 ¼0 ¼0 s0 s0 s0 ¼0 ¼0 s0 s0 s0 ¼0 ¼0 s0

2

3

[33] [33]

[33] [33]

[33] [33]

p 0.1

0.3

0.5

0.7

1.0

1.5

0.2196 0.2196 0.2062 0.2063 0.2142 0.4552 0.4552 0.4350 0.4351 0.4466 0.5514 0.5514 0.5288 0.5290 0.5418

0.2345 0.2345 0.2204 0.2205 0.2285 0.4867 0.4867 0.4649 0.4650 0.4773 0.5897 0.5896 0.5651 0.5653 0.5791

0.2503 0.2503 0.2355 0.2356 0.2438 0.5201 0.5200 0.4966 0.4968 0.5098 0.6302 0.6302 0.6037 0.6039 0.6187

0.2671 0.2671 0.2515 0.2516 0.2601 0.5555 0.5554 0.5303 0.5305 0.5443 0.6733 0.6733 0.6447 0.6449 0.6608

0.2944 0.2944 0.2774 0.2776 0.2866 0.6126 0.6126 0.5850 0.5852 0.6002 0.7427 0.7427 0.7112 0.7114 0.7289

0.3460 0.3460 0.3264 0.3266 0.3370 0.7201 0.7201 0.6881 0.6884 0.7058 0.8730 0.8730 0.8365 0.8368 0.8570

2.0

2.5

3.0

0.4065 0.3835 0.3838 0.3964

0.4775 0.4502 0.4504 0.4664

0.5603 0.5278 0.5281 0.5485

0.8449 0.8085 0.8088 0.8289

0.9898 0.9490 0.9493 0.9725

1.1580 1.1125 1.1129 1.1397

1.0240 0.9828 0.9832 1.0061

1.1990 1.1536 1.1540 1.1797

1.4017 1.3523 1.3528 1.3813

2.0

2.5

3.0

0.2263 0.2185 0.2242 0.2249

0.2162 0.2094 0.2140 0.2182

0.2045 0.1985 0.2023 0.2102

0.3621 0.3497 0.3588 0.3602

0.3460 0.3344 0.3425 0.3496

0.3273 0.3165 0.3237 0.3368

0.4074 0.3934 0.4036 0.4053

0.3893 0.3761 0.3854 0.3934

0.3683 0.3558 0.3642 0.3789


Table 7 Non-dimensional stress txz ð0Þ ¼ a hq0 txz 0; 2b; 0 of EGM plates subjected to sinusoidal distributed load (a/h ¼ 10). b/a

Theory

εz

1

Thai and Kim [32] Mantari and Soares Mantari and Soares Present Study Present Study Thai and Kim [32] Mantari and Soares Mantari and Soares Present Study Present Study Thai and Kim [32] Mantari and Soares Mantari and Soares Present Study Present Study

s0 s0 ¼0 ¼0 s0 s0 s0 ¼0 ¼0 s0 s0 s0 ¼0 ¼0 s0

2

3

[33] [33]

[33] [33]

[33] [33]

p 0.1

0.3

0.5

0.7

1.0

1.5

0.2454 0.2454 0.2380 0.2434 0.2367 0.3927 0.3927 0.3810 0.3896 0.3790 0.4418 0.4418 0.4286 0.4383 0.4265

0.2450 0.2450 0.2376 0.2430 0.2364 0.3920 0.3921 0.3803 0.3889 0.3787 0.4411 0.4411 0.4279 0.4376 0.4261

0.2442 0.2442 0.2368 0.2422 0.2359 0.3908 0.3908 0.3790 0.3877 0.3779 0.4396 0.4396 0.4264 0.4361 0.4252

0.2430 0.2430 0.2356 0.2410 0.2353 0.3889 0.3889 0.3770 0.3857 0.3768 0.4375 0.4375 0.4242 0.4340 0.4239

0.2405 0.2405 0.2330 0.2385 0.2338 0.3849 0.3849 0.3730 0.3817 0.3744 0.4330 0.4330 0.4196 0.4294 0.4212

0.2344 0.2344 0.2268 0.2324 0.2300 0.3752 0.3752 0.3630 0.3719 0.3684 0.4222 0.4221 0.4084 0.4185 0.4146

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

211

1.0

1.0 Present Study, E0/E1=1

0.9

0.9

Present Study, E0/E1=10

0.8

0.8

Present Study, E0/E1=0.1

0.7

0.6

Kashtalyan [1], E0/E1=10

0.6

0.5

Kashtalyan [1], E0/E1=0.1

z/h

Kashtalyan [1], E0/E1=1

z/h

0.7

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0 -4

-2

0

2

4

0.0

6

-2.5

-1.5

-0.5

σx

0.5

1.5

2.5

τxy 1.0

1.0

0.9

0.9 0.8

0.8

0.7

0.7

0.5

z/h

0.6

z/h

0.6

0.5 0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 -0.8

-0.6

0.0 -1.0

0.0 -0.1

-0.3

-0.8

-0.6

-0.4

-0.2

0.0

σz

τxz 1.0

1.0

0.9

0.9

0.8

0.8

0.7 0.7 0.6

0.5

z/h

z/h

0.6

0.4

0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0 -2.5

-1.5

-0.5

0.5

1.5

0.0 -5

u

-4

-3

-2

-1

0

w Fig. 4. Non-dimensional displacement and stresses through the thickness for sinusoidal loading of EGM plate.

The stress and displacement distributions through the thickness of Al/Al2O3 FGM square plate, under sinusoidal load, are presented in Fig. 3. The results are plotted as compared with the power series expansion of displacement components theory of Matsunaga [3] for various values of power law index k. According to Fig. 3, the

results are in good agreement with the ones of Matsunaga [3]. It is important to note that, the through the thickness distributions of in-plane stresses sx and txy are linear for homogeneous plate while it is parabolic for FGM plates. In Fig. 3, the non-dimensional quantities defined as follows;


a b  

; ;z sx Ec b Ec a b 2 2 ; ; z ; sx ðzÞ ¼ uðzÞ ¼ u 0; ; z ; wðzÞ ¼ w ; 2 2 2 q0 hq0 hq0  

a b b ; ;z sz txz 0; ; z txy ð0; 0; zÞ 2 2 2 sz ðzÞ ¼ ; txy ðzÞ ¼ ; txz ðzÞ ¼ q0 q0 q0

212

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

Table 8 Non-dimensional central displacement uð0Þ ¼ GðhÞ u=h q0 and in-plane normal stress sx ð0Þ ¼ sx ð0Þ=q0 of EGM plates subjected to uniformly distributed load. h/a

0.2

E0/Eh

Theory

w

Vaghefi et al. [5] BEM Vaghefi et al. [5] FEM Present Study (εz s 0) Vaghefi et al. [5] BEM Vaghefi et al. [5] FEM Present Study (εz s 0) Vaghefi et al. [5] BEM Vaghefi et al. [5] FEM Zhang et al. [9] Present Study (εz s 0) Vaghefi et al. [5] BEM Vaghefi et al. [5] FEM Zhang et al. [9] Present Study (εz s 0)

sx

0.3

w

sx

0.1

0.5

1

2

10

4.0916 4.1215 3.8333 15.356 15.403 16.322 0.9707 0.9732 0.9735 0.8923 7.2230 7.2639 7.1493 7.6576

8.9751 9.0047 8.8724 9.2902 9.2995 9.6545 2.1378 2.1407 2.1405 2.0834 4.3084 4.3378 4.3227 4.5062

12.599 12.613 12.597 7.4462 7.4588 7.6944 2.9853 2.9792 2.9795 2.9602 3.4496 3.4681 3.4710 3.5748

17.664 17.711 17.744 5.9410 5.9591 6.1109 4.1208 4.1333 4.1332 4.1669 2.7499 2.7673 2.7853 2.8235

39.060 39.155 38.333 3.4665 3.4805 3.4530 8.7134 8.7293 8.7343 8.9229 1.6449 1.6499 1.6759 1.5731

4.1.2. Exponentially graded plates In this section, the exponential function used to describe the material properties of the plate is given in Equation (2). The nondimensional stress and displacements of the plate are compared with the results of different shear deformation theories for different loadings. The non-dimensional displacements and stresses are presented in Tables 5e7 for various values of aspect ratio b/a, thickness ratio a/ h and exponent value p. Table 5 presents the central transverse displacements of the very thick EGM plates. The obtained results are compared with the quasi-3D sinusoidal and exact 3D elasticity theories of Zenkour [24], 2D and quasi-3D trigonometric theories of Mantari and Soares [33] and quasi-3D hyperbolic theory of Thai and Choi [35]. Since the proposed and other quasi-3D theories include the thickness-stretching effect, the results are close to each other. Meanwhile, 2D theories which do not include the thickness stretching effect overestimate the results. In Tables 6 and 7, the calculated non-dimensional stresses are presented as compared with the quasi-3D sinusoidal theory of Thai and Kim [32] and 2D and quasi-3D trigonometric theories of Mantari and Soares [33]. It is evident from the tables that the present computations are in an excellent agreement with the quasi-3D solutions of [32,33]. Tables 5e7 show also that transverse displacement w and transverse shear stress txz decrease and in plane stress sx increases with the increase of exponent p. In Fig. 4, distributions of non-dimensional displacements and stresses;


a b  

sx ; ;z G b G a b 2 2 uðzÞ ¼ 1 u 0; ;z ; wðzÞ ¼ 1 w ; ;z ; sx ðzÞ ¼ ; 2 2 2 q0 hq0 hq0  

a b b sz ; ;z txz 0; ;z txy ð0;0;zÞ 2 2 2 sz ðzÞ ¼ ; txy ðzÞ ¼ ; txz ðzÞ ¼ q0 q0 q0

through the thickness of an EGM plate under sinusoidal loading for different E0/E1 ratios are shown graphically (where; E1 ¼ E(h) and E0 ¼ E(0)). The results are compared with the 3D elasticity solution of Kashtalyan [1] (a/h ¼ 10). It can be seen from Fig. 4, the results of present quasi-3D theory are in excellent agreement with the 3D solution of Kashtalyan [1]. Table 8 presents the non-dimensional central transverse displacements and stresses of the EGM plates for different E0/Eh ratios. The obtained results are compared with the Finite Element Method (FEM) and Boundary Element Method (BEM) of Vaghefi et al. [5] and exact 3D elasticity theory of Zhang et al. [9] and they match very well.

4.2. Free vibration In this section various numerical examples are presented and discussed to verify the accuracy of the proposed new theories in predicting the free vibration responses of simply supported FG plates. As the first example, simply supported isotropic square plates are chosen in order to investigate the efficiency of the present theories. According to Eqs. (1)e(3), when the power law index p, approaches zero or infinity, the plate is isotropic composed of fully ceramic or metal, respectively. In Table 9, the first eight nondimensional natural frequencies are computed and compared with the results reported by the quasi-3D theories of Jha et al. [30] and Hebali et al. [34], exact three dimensional solution of Srinivas et al. [44] and first order shear deformation theory of Whitney and Pagano [45]. Table 9 shows that the computations based on the present theories are in excellent agreement with those predicted by the other quasi-3D theories of [30] and [34] for all modes of vibration. The next two examples are performed for Al/Al2O3 thick functionally graded square plates. In Table 10, non-dimensional fundamental frequencies of a square plate are computed for

Table 9 qffiffiffi Comparison of non-dimensional natural frequencies 6 ¼ uh Gr for isotropic square plate (a/h ¼ 10). Theory

εz

Jha et al. [30] Hebali et al. [34] Srinavas et al. [44] Whitney et al. [45] Present Study Present Study

s0 s0 s0 ¼0 ¼0 s0

Mode (m, n) (1,1)

(1,2)

(2,2)

(1,3)

(2,3)

(3,3)

(2,4)

(1,5)

0.0932 0.0933 0.0932 0.0930 0.0930 0.0932

0.2226 0.2228 0.2226 0.2220 0.2219 0.2227

0.3421 0.3422 0.3421 0.3406 0.3407 0.3424

0.4172 0.4173 0.4171 0.4149 0.4151 0.4176

0.5240 0.5240 0.5239 0.5206 0.5209 0.5247

0.6892 0.6890 0.6889 0.6834 0.6841 0.6902

0.7515 0.7512 0.7511 0.7447 0.7455 0.7526

0.9275 0.9268 0.9268 0.9174 0.9189 0.9290

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

213

Table 10 qffiffiffiffiffiffiffi Comparison of the non-dimensional fundamental frequencies 6 ¼ uh Ercc for Al/Al2O3 square plate. a/h

Theory

εz

2

Zhu and Liew [15] Matsunaga [25] Sheikholeslami and Saidi Belabed et al. [37] Present Study Present Study Zhu and Liew [15] Benachour et al. [18] Hosseini et al. [19] Matsunaga [25] Sheikholeslami and Saidi Belabed et al. [37] Present Study Present Study Zhu and Liew [15] Benachour et al. [18] Hosseini et al. [19] Matsunaga [25] Sheikholeslami and Saidi Belabed et al. [37] Present Study Present Study Benachour et al. [18] Hosseini et al. [19] Sheikholeslami and Saidi Belabed et al. [37] Present Study Present Study

¼0 s0 s0 s0 ¼0 s0 ¼0 ¼0 ¼0 s0 s0 s0 ¼0 s0 ¼0 ¼0 ¼0 s0 s0 s0 ¼0 s0 ¼0 ¼0 s0 s0 ¼0 s0

5

10

20

[31]

[31]

[31]

[31]

k 0

0.5

1

4

10

0.9265 0.9400 0.9400 0.9414 0.9303 0.9440 0.2111 0.2112 0.2113 0.2121 0.2121 0.2121 0.2113 0.2124 0.0576 0.0576 0.0577 0.0577 0.0577 0.0578 0.0577 0.0578 0.0148 0.0148 0.0148 0.0148 0.0148 0.0148

0.8060 0.8232 0.8223 0.8248 0.8115 0.8269 0.1804 0.1806 0.1807 0.1819 0.1818 0.1819 0.1807 0.1827 0.0489 0.0490 0.0490 0.0491 0.0491 0.0494 0.0490 0.0494 0.0125 0.0125 0.0125 0.0126 0.0125 0.0126

0.7331 0.7477 0.7475 0.7516 0.7360 0.7536 0.1629 0.1628 0.1631 0.1640 0.1640 0.1640 0.1631 0.1661 0.0441 0.0441 0.0442 0.0442 0.0442 0.0449 0.0442 0.0449 0.0113 0.0113 0.0113 0.0115 0.0113 0.0115

0.6112 0.5997 0.5995 0.6056 0.5921 0.6063 0.1395 0.1375 0.1378 0.1383 0.1382 0.1383 0.1378 0.1410 0.0381 0.0380 0.0381 0.0381 0.0381 0.0389 0.0380 0.0389 0.0098 0.0098 0.0098 0.0100 0.0098 0.0100

0.5640 0.5460 0.5461 0.5495 0.5413 0.5506 0.1323 0.1300 0.1301 0.1306 0.1306 0.1306 0.1300 0.1319 0.0365 0.0363 0.0364 0.0364 0.0364 0.0368 0.0363 0.0368 0.0094 0.0094 0.0094 0.0095 0.0094 0.0095

Table 11 pffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the first four non-dimensional natural frequencies 6 ¼ ua2 =h rc =Ec for Al/Al2O3 square plate (a/h ¼ 10). Mode (m, n)

Theory

εz

(1,1)

Benachour et al. [18] Matsunaga [25] Belabed et al. [37] Present Study Present Study Benachour et al. [18] Matsunaga [25] Belabed et al. [37] Present Study Present Study Benachour et al. [18] Matsunaga [25] Belabed et al. [37] Present Study Present Study

¼0 s0 s0 ¼0 s0 ¼0 s0 s0 ¼0 s0 ¼0 s0 s0 ¼0 s0

(1,2)

(2,2)

k 0

0.5

1

4

10

5.7690 5.7777 5.7800 5.7695 5.7807 13.760 13.810 13.800 13.765 13.817 21.125 21.210 21.210 21.127 21.237

4.9000 4.9170 4.9400 4.9015 4.9410 11.731 11.800 11.840 11.739 11.851 18.055 18.190 18.250 18.073 18.268

4.4160 4.4270 4.4900 4.4193 4.4907 10.576 10.630 10.770 10.590 10.773 16.282 16.400 16.590 16.313 16.609

3.8040 3.8110 3.8900 3.8064 3.8934 9.0120 9.0450 9.2300 9.0224 9.2314 13.756 13.830 14.090 13.777 14.099

3.6350 3.6420 3.6800 3.6365 3.6827 8.5570 8.5880 8.6800 8.5613 8.6768 12.995 13.060 13.180 13.002 13.186

Table 12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Comparison of non-dimensional fundamental frequencies 6 ¼ ua2 =h rM =EM for Al/ZrO2 square plate (m ¼ n ¼ 1) (Mori-Tanaka Homogenization). Theory

Benachour et al. [18] Matsunaga [25] Neves et al. [27] Belabed et al. [37] Alijani et al. [38] Vel et al. [43] Present Study Present Study

εz

¼0 s0 s0 s0 s0 s0 ¼0 s0

k¼0

k¼1

a/h ¼ 5

pffiffiffiffiffiffi a=h ¼ 10

a/h ¼ 10

a/h ¼ 5

a/h ¼ 10

a/h ¼ 20

k¼2

k¼3

k¼5

4.6220 4.6582

5.7600 5.7769

4.6591 4.6606 4.6582 4.6236 4.6672

5.7800 5.7769 5.7769 5.7695 5.7807

5.6750 5.7123 5.4825 5.4800 5.4796 5.4806 5.4216 5.4829

6.1800 6.1932 5.9600 5.9700 5.9578 5.9609 5.9136 5.9676

6.3200 6.3390 6.1200 6.1200 6.1040 6.1076 6.0647 6.1160

5.6225 5.6599 5.4950 5.5025 5.4919 5.4923 5.4431 5.5064

5.6375 5.6757 5.5300 5.5350 5.5279 5.5285 5.4798 5.5388

5.6650 5.7020 5.5625 5.5625 5.5633 5.5632 5.5138 5.5644

214

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

different power law index and different a/h raitos and compared with first order shear deformation theory of Zhu and Liew [15], 2D shear deformation theory of Benachour et al. [18] and quasi-3D shear deformation theories of Hosseini et al. [19], Matsunaga [25], Sheikholeslami and Saidi [31] and Belabed et al. [37]. Again, the obtained results are found to correlate exceptionally well with the other quasi-3D solutions, even for very thick plates. The table shows that, fundamental frequencies increase with the increase in the thickness of plate and decrease with the increase of power law index. In Table 11, to verify the higher order modes for functionally graded plates, the first three frequencies of the Al/Al2O3 FG square plates are computed and compared with the higher order shear deformation theory of Benachour et al. [18] and quasi-3D shear deformation theories of Matsunaga [25] and Belabed et al. [37]. As it is seen from the table, the present theories are in good agreement with those reported by the other quasi-3D theories of [25] and [37], particularly at the higher modes of vibration. It is seen from the tables that when the effects of normal deformations are neglected, the natural frequencies of functionally graded plates are found lower. Table 12 shows non-dimensional fundamental frequencies of Al/ ZrO2 functionally graded square plates. In order to compare results, we use the MorieTanaka scheme to describe the material properties of the FG plate which is given in Equation (3). The results of present theory are compared with the results of the higher order shear deformation theory of Benachour et al. [18] and quasi-3D shear deformation theory of Matsunaga [25], Neves et al. [27], Belabed et al. [37], Alijani and Amabili [38] and three dimensional exact solution of Vel and Batra [43]. It can be seen from Table 12 that, the results of present quasi-3D theory are in good agreement with the results of other quasi-3D theories. The small difference between the present 2D and quasi-3D shear deformation theory results is due to the neglecting the thickness stretching effect. 5. Conclusions A higher order enhanced theory has been presented for quasi3D static and free vibration analysis of thick FGM plates. The present method is based on the three dimensional elasticity theory and takes into account the effect of not only transverse shear strains but also transverse normal strains. The governing equations have been obtained by the Hamilton principle. Double Fourier series have been used to solve the partial differential equations. The accuracy of present method has been shown via the results obtained by present method compared with the results of the other theories. The results obtained by the present method can be summarized as follows:  Through all the comparative analyzes, it can be observed that the present theory shows good agreement with that of the results of other 2D and quasi-3D shear deformation theories. Also, the results predicted by the proposed theory are in an excellent agreement with 3D elasticity solutions even for the case of very thick plates.  The results show that the 2D and quasi-3D shear deformation theories have almost identical results for thin plates. For the thick and moderately thick plates, however, it has been seen from the comparison studies with the exact elasticity solutions that the quasi-3D theories which account for the transverse normal strain effects, can predict the static and free vibration behavior more accurately compared to other HDST theories. So, it is relevant to conclude that the effect of transverse normal strains on bending and free vibration behavior of composite plates are just as important as the effect of transverse shear strains and must be taken into account.

 Although the transverse stress components can be calculated from the constitutive equations, these stresses may not satisfy the stress boundary conditions on the top and bottom surfaces of the plate. So, the transverse stress components may be obtained by using equilibrium equations of three-dimensional elasticity theory as satisfying the stress boundary conditions.  The natural frequencies of plate decrease with the increase of power law index. Although increasing value of power law index causes to decrease in the natural frequency, the effect of the value of power law index more than 5 is negligible.  The small difference between the present 2D and quasi-3D shear deformation theory results is due to the neglecting the thickness stretching effect. If the effects of normal deformations neglected, the natural frequencies of functionally graded plates are found lower. Appendix The stress and moment resultants which appeared in Equation (8) are given by

8 9 3 fεg > > > Dij > > < fkg > = 7 Gij 5 ði ¼ 1; 3 j ¼ 1; 4Þ; > > > > fkq g > > Fij Hij Kij : ; fqg fRg ¼ Lij fkq g ði; j ¼ 1; 2Þ

9 2 8 Aij > = < fNg > 6 fMg ¼ 4 Bij > > ; : Cij fPg

Bij Eij

Cij Fij

(A1) 8 9 Mij >T > > > > > < Nij =   u0;x v0;y 0 w0;xx w0;yy 0 qx;x qy;y 0 0 0 qz T Sz ¼ > > O ij > > > > : ; Pij ði ¼ 1 j ¼ 1;3Þ (A2) 9 8 9 8 9 8   < Mx = < Px = < Nx = Rx Ny ; fMg ¼ My ; fPg ¼ Py ; fRg ¼ ; fNg ¼ Ry ; : ; : ; : Nxy Mxy Pxy (A3) 8 > > > <

9 9 8 w0;xx > > > > > > > > > = < v0;y w0;yy = ; fkg ¼  ; fεg ¼ > > > 0 0 > > > > > > > > > : ; ; : u0;y þ v0;x 2w0;xy 8 9 8 9 qx;x 0> > > > > > > > > > > < =   q þq  = <0> qy;y x z;x ; Kq ¼ ; fqg ¼ fkq g ¼ > > > > 0 qy þ qz;y q z > > > > > > > : ; ; : > qx;y þ qy;x 0 u0;x

(A4) where the stiffness components and inertias are given as:



Aij ;Bij ;Cij ;Dij ;Eij ;Fij ;Gij ;Hij ;Kij

¼

Zh=2 n

00

 00

00

1;z;f ðzÞ;f ðzÞ;z2 ;zf ðzÞ;zf ðzÞ;½f ðzÞ2 ;f ðzÞf ðzÞ

h=2

½Q1 ij dz i ¼ 1;3 j ¼ 1;4

o (A5)

S.S. Akavci, A.H. Tanrikulu / Composites Part B 83 (2015) 203e215

  Mij ;Nij ;Oij ;Pij Zh=2 n o 00 00 00 00 00 f ðzÞ; zf ðzÞ;f ðzÞ f ðzÞ; f ðzÞ f ðzÞ ½Q3 Tij dz i ¼ 1 j ¼ 1;3

¼

h=2

(A6)

  Lij ¼

8 > Zh=2 > > < > > h=2 > : 2

Q11 6 ½Q1  ¼ 4 Q12

2

½f 0 ðzÞ ½Q2 ij dz if i ¼ j 0 Q12 Q11

0 0 9 8 Q > > < 12 = fQ3 g ¼ Q12 > > ; : Q11

i; j ¼ 1; 2

(A7)

if isj Q12 Q12 0

3 0 7 0 5; Q44

 ½Q2  ¼

Q55 0

0 ; Q66

(A8) I1 ; I2 ; I3 ; I4 ; I5 ; I6 ; I7 ; I8 Zh=2 ¼

  2 rðzÞ 1; z; z2 ; f ðzÞ; zf ðzÞ; ½f ðzÞ2 ; f 0 ðzÞ; ½f 0 ðzÞ dz

(A9)

h=2

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