Static response and free vibration analysis of FGM plates using higher order shear deformation...

Applied Mathematical Modelling 34 (2010) 3991–4011

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Applied Mathematical Modelling

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Static response and free vibration analysis of FGM plates using higherorder shear deformation theory

Mohammad Talha, B.N. Singh *

Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur 721 302, India

a r t i c l e i n f o

Article history:Received 14 July 2009Received in revised form 24 February 2010Accepted 30 March 2010Available online 18 April 2010

Keywords:Functionally graded materialHigher order shear deformation theoryFinite element methodIndependent field variables

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.03.034

* Corresponding author. Tel.: +91 3222 283026; fE-mail address: [email protected] (B.N

a b s t r a c t

Free vibration and static analysis of functionally graded material (FGM) plates are studiedusing higher order shear deformation theory with a special modification in the transversedisplacement in conjunction with finite element models. The mechanical properties of theplate are assumed to vary continuously in the thickness direction by a simple power-lawdistribution in terms of the volume fractions of the constituents. The fundamental equa-tions for FGM plates are derived using variational approach by considering traction freeboundary conditions on the top and bottom faces of the plate. Results have been obtainedby employing a continuous isoparametric Lagrangian finite element with 13 degrees offreedom per node. Convergence tests and comparison studies have been carried out todemonstrate the efficiency of the present model. Numerical results for different thicknessratios, aspect ratios and volume fraction index with different boundary conditions havebeen presented. It is observed that the natural frequency parameter increases for plateaspect ratio, lower volume fraction index n and smaller thickness ratios. It is also observedthat the effect of thickness ratio on the frequency of a plate is independent of the volumefraction index. For a given thickness ratio non-dimensional deflection increases as the vol-ume fraction index increases. It is concluded that the gradient in the material propertiesplays a vital role in determining the response of the FGM plates.

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1. Introduction

Composite materials have been successfully used in aircraft and other engineering applications for many years because oftheir excellent strength to weight and stiffness to weight ratios. Recently, advanced composite materials known as function-ally graded material have attracted much attention in many engineering applications due to their advantages of being able toresist high temperature gradient while maintaining structural integrity [1]. The functionally graded materials (FGMs) aremicroscopically inhomogeneous, in which the mechanical properties vary smoothly and continuously from one surface tothe other. They are usually made from a mixture of ceramics and metals to attain the significant requirement of materialproperties.

Due to the increased relevance of the FGMs structural components in the design of aerospace structures, their static andvibration characteristics have attracted the attention of many scientists in recent years. It is observed from the literature thatthe amount of such work carried out for isotropic plates are considerable, and limited literature is available on compositeplates. However, the literature on the analysis of the FGMs plate is very few. Reddy [2] presented theoretical formulationand finite element models based on third order shear deformation theory for static and dynamic analysis of the FGM plates.

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3992 M. Talha, B.N. Singh / Applied Mathematical Modelling 34 (2010) 3991–4011

He obtained Navier solutions for a simply supported square plate under sinusoidally distributed load including the effect ofshear deformation.

Vel and Batra [3] presented a three-dimensional analytical solution for free and forced vibrations of simply supportedFGM rectangular plates. Roque et al. [4] performed the free vibration analysis of the FGM plates by using the multiquadricradial basis function method, i.e., meshless method and a higher order shear deformation theory (HSDT). Efraim and Eisen-berger [5] derived the equations of motion for annular plates using the first-order shear deformation theory (FSDT) whichincludes the effect of shear deformations to solve vibration frequencies and modes for various combinations of boundaryconditions.

Abrate [6] analyzed the problems of free vibrations, buckling, and static deflections of the FGM plates. The analysis ofthese problems is made using the classical laminated plate model, the FSDT model, and the HSDT model. Batra and Jin [7]used the first-order shear deformation theory coupled with the finite element method to study the free vibrations of rect-angular anisotropic FGM plate. Malekzadeh [8] presented the free vibration analysis of thick FGM plates supported on two-parameter elastic foundation.

Fazelzadeh et al. [9] investigated the applicability of differential quadrature method for vibration analysis of aero-ther-moelastic thin-walled blades made of FGM materials. First-order shear deformation theory is used to solve the governingequations of beams which include the effects of the rotary inertias and the blade presetting angle. Navazi and Haddadpour[10] analytically investigated the aero-thermoelastic stability margins of FGM panels in thermal environment. They em-ployed piston theory of aerodynamics to model quasi-steady aerodynamic loading.

Ng et al. [11] presented the parametric resonance of FGM rectangular plates under harmonic in-plane loading. The prob-lem is formulated using Hamiltons principle and the material properties are graded according to a simple power-law distri-bution in the thickness direction only. It is concluded that the origin of the instability regions can be easily controlled bycorrectly varying the power-law exponent. Reddy and Cheng [12] studied the harmonic vibration problem of FGM platesby means of a three-dimensional asymptotic approach, and the formulation is based in terms of transfer matrix. Nguyenet al. [13] have proposed the FSDT model in which transverse shear factors is obtained through these plate models by usingenergy equivalence methods.

Park [14] derived the frequency equation for the in-plane vibration of the clamped circular plate of uniform thicknesswith an isotropic material in the elastic range. Zenkour [15] studied the static response of a simply supported functionallygraded rectangular plate subjected to a transverse uniform load using the generalized shear deformation theory.

Uymaz and Aydogdu [16] presented three-dimensional vibration solutions for rectangular FGM plates with differentboundary conditions based on the small strain linear elasticity theory. Chebyshev polynomials based on Ritz energy ap-proach have been used in the analysis. Li et al. [17] studied free vibration of FGM sandwich rectangular plates with simplysupported and clamped edges, and the formulation is based on the three-dimensional elasticity theory. Ferreira et al. [18]have used collocation multiquadric radial basis functions to analyse static deformations of simply supported functionallygraded plate modelled by a higher order shear deformation theory.

The determination of accurate behavior of the FGMs largely depends on the theory used to model the structure because inthe FGMs, material properties vary continuously as a function of position in the preferred direction. Various concepts havebeen developed to inculcate the appropriate analysis of the FGM plates. The classical Kirchhoff plate theory neglects trans-verse shear deformation and gives acceptable results for relatively thin plates. In order to circumvent this problem an earlierattempts were made by Reissner [19] and Mindlin [20]. However, a shear-correction factor must be incorporated to over-come the problem of a constant transverse shear stress distribution and its value depends on various parameters, such asapplied loads, boundary conditions and geometric parameters, etc. The inaccuracy occurs due to neglecting the effects oftransverse shear and normal strains in the plate [21]. Due to continuous variation in material properties, the first-order sheardeformation theory and higher order shear deformation theory may be conveniently used in the analysis. It is noted that thefirst-order shear deformation theory proposed by Mindlin [20] does not satisfy the parabolic variation of transverse shearstrain in the thickness direction. Subsequently, many higher order theories were proposed, notable among them are [22–25]. The higher order theories assume the in-plane displacements as a cubic expression of the thickness coordinate andthe out-of-plane displacement to be constant. Thus, the development of higher order shear deformation theory to assimilatethe behaviour of FGM structures has been of high importance to the researchers.

Therefore, keeping this view point in mind the present study aims to develop a higher order shear deformation theorywith a special modification in the transverse displacement which contributes additional freedom to the displacementsthrough the thickness and fundamentally eradicates the over-correction. To implement this theory a suitable C0 continuousisoparametric finite element with 13 degrees of freedom (DOFs) per node is proposed in order to reduce the computationalefforts required in the formation of element matrices without affecting the solution accuracy. It is assumed that the plate ismade of an isotropic material and its properties varying only through the thickness direction according to a simple power-law function of the position. Numerical results have been presented for ceramic–metal graded plates with different bound-ary and loading conditions. The convergence and comparison studies manifest the accuracy and precision of the presentproposed method. The effects of aspect and thickness ratios, volume fraction index, various boundary and loading conditionson free vibration and static responses are investigated. Some results are presented in the form of tables and figures, whichcan suit as a benchmark for the future research.

The paper is organised as follows. Section 2 gives the brief description of geometric configuration and material propertiesof the plate. Theoretical formulations of the problem are given in Section 3. In Section 4 solution methodology is presented

M. Talha, B.N. Singh / Applied Mathematical Modelling 34 (2010) 3991–4011 3993

followed by the convergence and validation studies in Section 5. Section 6 explains the results and discussion and Section 7accomplishes the conclusions.

2. Geometric configuration and material properties

Here we consider an FGM plate of length a, width b and total thickness h made from a mixture of metal and ceramics, inwhich the composition is varied from the top to the bottom surface as shown in Fig. 1. The top surface (Z = h/2) of the plate isceramic-rich, whereas the bottom surface (Z = �h/2) is metal rich. All formulation are confined here with the assumption of alinear elastic material behavior and small displacements and strains.

The elastic material properties vary through the plate thickness according to the volume fractions of the constituents.Power-law distribution is commonly used to describe the variation of material properties (Fig. 2), which is expressed as

PðzÞ ¼ ðPc � PmÞV c þ Pm; ð1Þ

VcðzÞ ¼zhþ 1

2

� �n

ð0 6 n 61Þ; ð2Þ

where P denotes the effective material property, Pm and Pc represents the properties of the metal and ceramic, respectively,Vc is the volume fraction of the ceramic and n is the volume fraction exponent. The effective material properties of the plate,including Young’s modulus E, density q vary according to Eq. (1) and m is assumed to be constant.

3. Theoretical formulation

3.1. Displacement field and strains

In the present study, system of governing equations for FGM plate is derived by using variational approach. The origin ofthe material coordinates is at the middle of the plate as shown in Fig. 1. For accurate analysis of transverse shear effects inthe mathematical formulation the HSDT model has been used with a special modification in the transverse displacement.The in-plane displacements �u; �v and the transverse displacement �w for the plate are assumed as,

�uðx; y; z; tÞ ¼ u0ðx; y; tÞ þ zwxðx; y; tÞ þ z2nxðx; y; tÞ þ z3qxðx; y; tÞ;

�vðx; y; z; tÞ ¼ v0ðx; y; tÞ þ zwyðx; y; tÞ þ z2nyðx; y; tÞ þ z3qyðx; y; tÞ;

�wðx; y; z; tÞ ¼ w0ðx; y; tÞ þ zwzðx; y; tÞ þ z2nzðx; y; tÞ;

ð3Þ

where �u; �v and �w denote the displacements of a point along the (x,y,z) coordinates. u0, v0, and w0 are corresponding dis-placements of a point on the mid-plane. wx and wy are the rotations of normal to the mid-plane about the y-axis and x-axis,respectively. The functions nx, ny, qx and qy are the higher order terms in the Taylor series expansion defined in the mid-planeof the plate. The higher order terms are determined by vanishing the transverse shear stresses sxz = s4 and syz = s5 on the topand bottom surfaces of the plate and by applying this boundary condition the displacement field is modified as

�u ¼ u0 þ f1ðzÞwx þ f2ðzÞax þ f3ðzÞbx þ f4ðzÞhx;

�v ¼ v0 þ f1ðzÞwy þ f2ðzÞay þ f3ðzÞby þ f4ðzÞhy;

�w ¼ w0 þ f5ðzÞwz þ f6ðzÞaz;

ð4Þ

a

b x

yz

2h

2h

Fig. 1. Geometry and dimensions of the plate.

Fig. 2. Variation of the volume fractions Vc through the thickness.


where f1(z) = C1z � C2z3, f2(z) = �C3z2, f3(z) = �C4z3, f4(z) = �C5z3, f5 = C1z, f6 = C1z2, C1 = 1, C2 = C4 = 4/3h2, C3 = 1/2, C5 = 1/3 andnz = az.

To assure the field variables are continuous within the element for C0 continuous finite element modelling, the out ofplane derivatives are replaced by incorporating the following relations in Eq. (4) as:

ax ¼owz

ox; bx ¼

owox

; hx ¼onz

ox; ay ¼

owz

oy; by ¼

owoy

; hy ¼onz

oy: ð5Þ

However, the above substitution (Eq. (5)) imposes an artificial constraints which are enforced variationally through a pen-alty approach [26] as:

owz

ox� ax ¼ 0;

owz

oy� ay ¼ 0;

owox� bx ¼ 0;

owoy� by ¼ 0;

onz

ox� hx ¼ 0

onz

oy� hy ¼ 0: ð6Þ

From Eq. (4), the field variables (basic unknowns) are interpreted as u, v, w, wx, wy, wz, ax, ay, az, bx, by, hx and hy for struc-tural deformation. Mathematically, it can be represented as

fKg ¼ u; v ; w; wx; wy; wz; ax; ay; az; bx; by; hx; hy� �T

; ð7Þ

where {K} is named as displacement vector.The linear strains corresponding to the displacement field as given in Eq. (4) is expressed as

��f g ¼ f�xx; �yy; �zz; cyz; cxz; cxygT; ð8Þ

or, it may be written as

exx

eyy

ezz

cyz

cxz

cxy

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

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

¼

e01

e02

e03

e04

e05

e06

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

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

þ z

k11

k12

k13

k14

k15

k16

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

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

þ z2

k21

k22

0

k24

k25

k26

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

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

þ z3

k31

k32

0

0

0

k36

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

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

; ð9Þ

where


e01 ¼

ou0

ox; e0

2 ¼ov0

oy; e0

3 ¼ wz; e04 ¼ wy þ

ow0

oy; e0

5 ¼ wx þow0

ox; e0

6 ¼ou0

oyþ ov0

ox;

k11 ¼

owx

ox; k1

2 ¼owy

oy; k1

3 ¼ 2az; k14 ¼

owz

oy� ay; k1

5 ¼owz

ox� ax; k1

6 ¼owx

oyþ

owy

ox;

k21 ¼ �C3

owx

ox; k2

2 ¼ �C3oay

oy; k2

4 ¼oaz

oy� hy � 3C2ðwy þ byÞ; k2

5 ¼oaz

ox� hx � 3C2ðwx þ bxÞ;

k26 ¼ �C3

oax

oyþ oay

ox

� �; k3

1 ¼ �C2owx

oxþ obx

ox

� �� C5

ohx

ox; k3

2 ¼ �C2owy

oyþ

oby

oy

� �� C5

ohy

oy;

k36 ¼ �C2

owx

oyþ

owy

oxþ obx

oyþ

oby

ox

� �� C5

ohx

oyþ ohy

ox

� �:

3.2. Constitutive relations

The linear constitutive relations are [27]

rxx

ryy

rzz

ryz

rxz

rxy

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

9>>>>>>>>=>>>>>>>>;¼

Q11 Q12 Q 13 0 0 0Q12 Q22 Q 23 0 0 0Q13 Q23 Q 33 0 0 0

0 0 0 Q 44 0 00 0 0 0 Q 55 00 0 0 0 0 Q 66

2666666664

3777777775

exx

eyy

ezz

cyz

cxz

cxy

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

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

; ð10Þ

where

Q 11 ¼ Q 22 ¼ Q 33 ¼EðzÞð1� m2Þð1� 3m2 � 2m3Þ ;

Q 12 ¼ Q 13 ¼ Q 23 ¼EðzÞmð1þ mÞð1� 3m2 � 2m3Þ ;

Q 44 ¼ Q 55 ¼ Q 66 ¼EðzÞ

2ð1þ mÞ :

ð11Þ

The modulus E and the elastic coefficients Qij vary through the plate thickness according to Eqs. (1) and (2).

3.3. Strain energy

The strain energy of the FGM plate is given by,

U ¼ 12

ZvfegT

i frgi dV : ð12Þ

3.4. Kinetic energy

The kinetic energy of the FGM plate can be expressed as

T ¼ 12

ZVq �_u� �T �_u

� �dV ; ð13Þ

where q and �uf g are the density and global displacement vector of the plate. The global displacement field model as given byEq. (4) may be represented as,

�uf g ¼ N� �fKg; ð14Þ

where {K} is as defined in Eq. (7) and the function of thickness coordinate N� �

is defined as

N� �¼

1 0 0 f1ðzÞ 0 0 f2ðzÞ 0 0 f3ðzÞ 0 f4ðzÞ 00 1 0 0 f1ðzÞ 0 0 f2ðzÞ 0 0 f3ðzÞ 0 f4ðzÞ0 0 1 0 0 f5ðzÞ 0 0 f6ðzÞ 0 0 0 0

264

375; ð15Þ

by substituting Eq. (14) into Eq. (13), the kinetic energy becomes

T ¼ 12

ZA

ZZq _Kn oT

N� �T

N� �

_Kn o

dz� �

dA ¼ 12

ZA

_Kn oT

½m� _Kn o

dA: ð16Þ

3

6

9

2 1

4 5

7 8

Fig. 3. Node number for nine-noded rectangular element in natural coordinate system.

Table 1Properties of the FGM components.

Material Properties

E (N/m2) m q (kg/m3)

Aluminium (Al) 70 � 109 0.30 2707Alumina (Al2O3) 380 � 109 0.30 3800Zirconia (ZrO2) 151 � 109 0.30 3000Silicon carbide (SiC) 427 � 109 0.17 3210Ti–6Al–4V 105.7 � 109 0.298 4429Stainless steel (SUS304) 207.78 � 109 0.3177 8166Silicon nitride (Si3N4) 322.27 � 109 0.24 2370

Table 2Comparison of linear frequency �x ¼ x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð1� v2Þqcx2

l a2b2=p4Ech2

qfor various boundary conditions, n = 5 for (Al/ZrO2) square plate with different mess size.

Mesh size Boundary conditions

SSSS CCCC SCSC

Present (2 � 2) 1.4564 2.5371 2.1457Present (3 � 3) 1.4321 2.2669 2.0260Present (4 � 4) 1.4222 2.1944 1.9287Present (5 � 5) 1.4165 2.1540 1.8161Uymaz and Aydogdu [16] 1.4106 2.1447 1.8055

Percentage difference 0.416 0.431 0.583

Table 3Comparison of linear frequency �x ¼ xða2=hÞ qmð1� m2Þ=Em

� �1=2 with volume fraction index n of (Ti–6Al–4V/ZrO2) square plate for simply supported (SSSS)boundary condition, with different mess size.

Mesh size Volume fraction index (n)

ZrO2 0.5 1 2 Ti–6Al–4V

Present (2 � 2) 8.6061 7.4217 6.9653 6.5789 5.6446Present (3 � 3) 8.3961 7.2380 6.7907 6.4143 5.5066Present (4 � 4) 8.3245 7.1758 6.7319 6.3589 5.4596Present (5 � 5) 8.2877 7.1440 6.7022 6.3310 5.4102Huang and Shen [30] 8.273 7.139 6.657 6.286 5.400

Percentage difference 0.17 0.07 0.67 0.71 0.18

Table 4Comparison of linear frequency �x ¼ xa2=h qc=Ecð Þ1=2 with volume fraction index n of (SUS304/Si3N4) square plate for simply supported (SSSS) boundarycondition, with different mess size.


0 0.5 1 2 5 8 10

Present (2 � 2) 5.8914 4.1041 3.6913 3.2603 2.9539 2.8500 2.8099Present (3 � 3) 5.7841 4.0223 3.4910 3.1696 2.8728 2.7721 2.7331Present (4 � 4) 5.6769 3.9870 3.4894 3.1411 2.8472 2.7476 2.7089Present (5 � 5) 5.6523 3.9201 3.4415 3.1062 2.8210 2.7258 2.6973Zhao et al. [31] 5.6148 3.8947 3.4242 3.0813 2.8058 2.7129 2.6768

Percentage difference 0.663 0.647 0.502 0.801 0.538 0.473 0.760


Table 7Variation of the frequency parameter �x ¼ x


l a2b2=p4Ech2

qwith the volume fraction index n for (SSSS) square (Al/ZrO2) FGM plates (a/h = 10).

Boundary condition Mode n

Ceramic 0.5 1 5 10 Metal

SSSS 1 1.9362 1.7566 1.6853 1.5692 1.5189 1.40992 4.7231 4.2631 4.0609 3.7653 3.6664 3.43633 4.7231 4.2631 4.0609 3.7654 3.6666 3.43674 7.2780 6.5850 6.2752 5.7862 5.6308 5.29215 9.1111 8.2711 7.8766 7.0493 6.8266 6.5701

Table 6Variation of the frequency parameter �x ¼ x


l a2b2=p4Ech2

qwith the volume fraction index n for (CCCC) square (Al/ZrO2) FGM plates (a/h = 10).



CCCC 1 3.4173 3.0897 2.9436 2.7191 2.6461 2.48492 6.7029 6.0788 5.7875 5.2956 5.1529 4.86823 6.7029 6.0788 5.7875 5.2956 5.1529 4.86824 9.4603 8.5930 8.1787 7.4474 7.2463 6.86705 11.6199 10.5751 10.0627 9.1182 8.8667 8.4264

Fig. 4. Comparison of the computed non-dimensional center deflection of an Al/SiC plate with [32].

Table 5Comparison of the centroidal deflection of a simply supported (SSSS) (Al/ZrO2) square plate for various volume fraction index n and a/h = 5, with different messsize.


Ceramic 0.5 1 2 Metal

Present (2 � 2) 0.0285 0.0357 0.0394 0.0430 0.0596Present (3 � 3) 0.0352 0.0439 0.0487 0.0536 0.0738Present (4 � 4) 0.0269 0.0336 0.0373 0.0409 0.0564Present (5 � 5) 0.0250 0.0319 0.0358 0.0393 0.0541Ferreira et al. [18] 0.0247 0.0313 0.0351 0.0388 0.0534

Percentage difference 1.20 1.88 1.81 1.27 1.29



3.5. Work done due to transverse load

The external work done on the plate by uniformly applied load p0 may be written as

Table 1Variatio

Boun

CCCC

Table 1Variatio

Boun

SSSS

Table 1Variatio

Boun

CCCC

Table 9Variatio

Boun

SSSS

Table 8Variatio

Boun

CCCC

Wext ¼12

ZA

p0ðx; yÞfwgdA: ð17Þ

2n of the frequency parameter �x ¼ xa2=h

ffiffiffiffiffiffiffiffiffiffiffiffiqc=Ec

pwith the volume fraction index n for (CCCC) square (SUS304/Si3N4) FGM plates (a/h = 20).

dary condition Mode n


1 11.0932 7.6528 6.7074 5.4592 5.1991 4.85132 23.2389 16.0474 14.0622 11.4138 10.8706 10.15983 23.2389 16.0474 14.0622 11.4138 10.8706 10.15984 33.8879 23.4088 20.5156 16.6363 15.8439 14.81465 44.3223 30.6566 26.8554 21.7155 20.6790 19.3671



pwith the volume fraction index n for (SSSS) square (SUS304/Si3N4) FGM plates (a/h = 10).



1 5.7523 3.9701 3.4845 2.8351 2.6973 2.51542 14.0336 9.6890 8.4903 6.8941 6.5669 6.13613 14.0354 9.6906 8.4918 6.8952 6.5680 6.13704 21.6188 14.9404 13.0959 10.6102 10.1053 9.45155 27.1449 18.7691 16.4526 13.3057 12.6668 11.8664



pwith the volume fraction index n for (CCCC) square (SUS304/Si3N4) FGM plates (a/h = 10).



1 10.1599 7.0202 6.1489 4.9816 4.7457 4.44102 19.9367 13.7978 12.0812 9.7440 9.2841 8.71073 19.9367 13.7978 12.0812 9.7440 9.2841 8.71074 28.1367 19.4845 17.0625 13.7350 13.0873 12.29195 34.6017 23.9945 20.9992 16.8507 16.0556 15.1084

n of the frequency parameter �x ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð1� v2Þqcx2

l a2b2=p4Ech2

qwith the volume fraction index n for (SSSS) square (Al/ZrO2) FGM plates (a/h = 20).



1 1.9943 1.8074 1.7348 1.6215 1.5693 1.45302 5.0388 4.5415 4.3322 4.0467 3.9372 3.66953 5.0392 4.5417 4.3322 4.0467 3.9375 3.66984 7.9752 7.1971 6.8699 6.4057 6.2286 5.80725 10.3318 9.3352 8.9029 8.2691 8.0441 7.5198

n of the frequency parameter �x ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð1� v2Þqcx2

l a2b2=p4Ech2

qwith the volume fraction index n for (CCCC) square (Al/ZrO2) FGM plates (a/h = 20).



1 3.7343 3.3670 3.2102 2.9929 2.9124 2.71852 7.8219 7.0664 6.7349 6.2424 6.0737 5.68993 7.8219 7.0664 6.7349 6.2424 6.0737 5.68994 11.4172 10.3233 9.8377 9.0950 8.8488 8.30295 14.9035 13.5009 12.8641 11.8399 11.5115 10.8283


4. Solution methodology

4.1. Finite element model

4.1.1. IntroductionFrom Eq. (4), it is seen that the expressions for in-plane displacement �u and �v involve the derivatives of out-of-plane dis-

placement w0. As a result of this, second order derivatives would be present in the strain vector, thus necessitating theemployment of C1 continuity for finite element analysis. The complexity and difficulty involved with making a choice ofC1 continuity is well known, and C0 continuity permits easy isoparametric finite element formulation and consequentlycan be applied for non-rectangular geometries as well. Here in the present study, the derivatives are assumed as independentfield variables by treating ax, ay, bx, by, hx and hy as a separate DOFs. In this process, DOFs increase from 11 to 13 for the HSDT.However, the strain vector will be having only the first-order derivatives, and hence a C0 continuous element would be suf-ficient for the present model. However, artificial constraints are imposed which can be enforced variationally through a pen-alty approach in order to satisfy the constraints emphasized in Eq. (6).

Fig. 5. Variation of frequency parameter with volume fraction index n and a/h ratio for clamped (CCCC) Al/Al2O3 square plate.

Table 14Non-dimensional center deflection with aspect ratio (b/a) for clamped-free (CFCF) Al/ZrO2 FGM plate.

Aspect ratio (b/a) wc

n = 0 n = 0.5 n = 1 n = 2 n = 5 n = 10

0.5 0.1153 0.1447 0.1626 0.1814 0.2026 0.21581.0 0.5189 0.6556 0.7352 0.8135 0.8966 0.95451.5 0.9742 1.2343 1.3832 1.5254 1.6724 1.78042.0 1.3412 1.7016 1.9062 2.0984 2.2948 2.44302.5 1.5923 2.0217 2.2644 2.4904 2.7196 2.89513.0 1.7557 2.2301 2.4976 2.7452 2.9953 3.1886

Table 13Variation of the frequency parameter �x ¼ xa2=h


pwith the volume fraction index n for (SSSS) square (SUS304/Si3N4) FGM plates (a/h = 20).



SSSS 1 5.9240 4.0853 3.5865 2.9237 2.7815 2.59102 14.9636 10.3158 9.0439 7.3701 7.0188 6.54493 14.9651 10.3158 9.0439 7.3712 7.0198 6.54564 23.6605 16.3181 14.3108 11.6551 11.0973 10.34875 30.6973 21.1826 18.5757 15.1021 14.3809 13.4247


A nine-noded isoparametric element is employed for finite element modeling. In the finite element, the domain is discret-ized into a set of finite elements. Over each of the elements, the displacement vector and element geometry of the model isexpressed by

fKg ¼XNN

i¼1

NifKgi; x ¼XNN

i¼1

Nixi; y ¼XNN

i¼1

Niyi; ð18Þ

where Ni is the interpolation function (shape function) for the ith node, {K}i is the vector of unknown displacements for theith node, NN is the number of nodes per element and xi and yi are Cartesian coordinate of the ith node. The shape functionswhich are used in the finite element approximation are mentioned below [28] and the node numbering are shown in Fig. 3.

N1 ¼14

n2 � n

g2 � g

; N2 ¼12

1� n2 g2 � g

;

N3 ¼14

n2 þ n

g2 � g

; N4 ¼12

n2 � n

1� g2

;

N5 ¼ 1� n2 1� g2

; N6 ¼12

n2 þ n

1� g2 ;

N7 ¼14

n2 � n

g2 þ g

; N8 ¼12

1� n2 g2 þ g

;

N9 ¼14

n2 þ n

g2 þ g

:

4.1.2. Strain energy of the plateThe strain energy of the FGM plate is given by,

U ¼XNE

e¼1

UðeÞ: ð19Þ

Here NE is number of elements used for messing the plate U(e) is the elemental strain energy which can be obtained usingEqs. (10), (12) and (14) and expressed as

U ¼ 12

XNE

e¼1

fKgeðTÞ½K�ðeÞfKgðeÞ: ð20Þ

Here [K](e) and {K}(e) are defined as linear stiffness matrix and displacement vector for the eth element, respectively.

4.1.3. Kinetic energy of the plateSumming over total number of element NE, kinetic energy of vibrating plate may be written as T ¼

PNNe¼1T ðeÞ, where

Fig. 6. Variation of frequency parameter with volume fraction index n and a/h ratio for clamped (CCCC) Al/Al2O3 rectangular plate.

F


TðeÞ ¼ 12

ZAðeÞ

_Kn oðeÞT

½m�ðeÞ _Kn oðeÞ

dA: ð21Þ

Here [m](e) is the inertia matrix of element.

4.1.4. Work done due to external transverse loadThe work done by external uniformly applied mechanical load p0(x,y) is given by

V ¼ �Wext ¼12

ZA

p0ðx; yÞwdA: ð22Þ

Using the finite element notation model, above equation may be written as

V ¼XNE

e¼1

V ðeÞ; ð23Þ

where V ðeÞ ¼ �R

AðeÞ fKgTfPgdA ¼ �fKgðeÞTfPgðeÞ with {P}(e) = (0 0 p0 0 0 0 0 0 0 0 0 0 0)T(e).

ig. 8. Variation of frequency parameter with volume fraction index n and a/h ratio for simply supported (SSSS) Al/Al2O3 rectangular plate.

Fig. 7. Variation of frequency parameter with volume fraction index n and a/h ratio for simply supported (SSSS) Al/Al2O3 square plate.


4.2. Governing equation

The governing equation for free vibration and static problem of FGM plate can be derived using variational principle,which is generalisation of the principle of virtual displacement. Lagrange equation for a conservative system can be writtenas [29]

Fig. 10.

Fig. 9

ddt

oTo _qif g

� �þ oU

ofqigþ o

ofqigc2

Z ax � owzox

2 þ bx � owzox

2

þ hx � onzox

2 þ ay � owzoy

� �2

þ by � owoy

� �2þ hy � onz

oy

� �2

2666664

3777775

dv

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

9>>>>>=>>>>>;� oV

ofqg ¼ 0; ð24Þ

for i = 1,2, . . ..Here {qi} and _qif g are the generalized coordinates and generalized velocities, respectively, and c is the penalty parameter

that enforces constraints, as stated in Section 3.1 (Eqs. (5) and (6)). The equilibrium equation for free vibration and static

Variation of frequency parameter with volume fraction index n and a/h ratio for simply supported and clamped (SSCC) Al/Al2O3 rectangular plate.

. Variation of frequency parameter with volume fraction index n and a/h ratio for simply supported and clamped (SSCC) Al/Al2O3 square plate.


analysis with small deformation (i.e., small displacements and strain relation) can be obtained by substituting Eqs. (20), (21)and (23) in Eq. (24) as follows:

½M� €qf g þ K þ cKc½ �fqg ¼ fFg; ð25Þ

where [M], [K], [Kc] {q}, €qf g and {F} are global mass matrix, global linear stiffness matrix, global linear stiffness matrix arisingdue to constrains, global displacement, global acceleration and force vector, respectively.

The generalized governing equation (24) can be employed to study the free vibration and static analysis by dropping theappropriate terms as:

1. Generalized eigenvalue problem for a system vibrating in principle mode with natural frequency x, can be expressed as

K þ cKc½ �fqg ¼ k½M�fqg; ð26Þ

with k = x2, where x is defined as frequency of natural vibration.2. Linear static analysis

K þ cKc½ �fqg ¼ fFg: ð27Þ

Fig. 12. Non-dimensional center deflection vs. volume fraction index n for simply supported plate (SSSS) Al/Al2O3 square plate.

Fig. 11. Non-dimensional center deflection vs. volume fraction index n for clamped (CCCC) Al/Al2O3 square plate.


5. Numerical results

The free vibration and static responses of the FGMs plate are computed using the proposed mathematical model in con-junction with FEM. A computer programme has been developed in MATLAB 7.5.0 (R2007b) environment. The validation andefficacy of the proposed algorithm is examined by comparing the results with those available in the literature. A nine-nodedLagrange isoparametric element, with 117 degrees of freedom per element for the present HSDT model has been used fordiscretizing the plate. For the computation of results full integration schemes (3 � 3) are used for thick plates and selectiveintegration schemes (2 � 2) for thin plates.

In the present analysis, various boundary conditions are used to check the efficacy of the model. However, the formula-tion and code do not put any limitations. Table 1 shows the properties of the FGMs constituents given at room temperature(300 K), which have been used for the computation of the results throughout the study, unless specified otherwise. Fig. 2shows the volume fraction of the ceramic phase through the dimensionless thickness. It is assumed that the materials areperfectly elastic throughout the deformation. The boundary conditions used in the analysis are as follows:

Simply supported (SSSS):u0 = w0 = wy = ax = az = by = hx = 0, at x = 0 and a.v0 = w0 = wx = ay = az = bx = hy = 0, at y = 0 and b.

Fig. 14. Non-dimensional center deflection vs. volume fraction index n for clamped (CCCC) Al/Al2O3 rectangular plate.

Fig. 13. Non-dimensional center deflection vs. volume fraction index n for simply supported and clamped (SSCC) Al/Al2O3 square plate.


Clamped (CCCC):u0 = v0 = w0 = wx = wy = wz = ax = ay = az = bx = by = hx = hy = 0, at x = 0, a and y = 0, b.Simply supported and clamped (SSCC):u0 = w0 = wy = ax = az = by = hx = 0, at x = 0 and a.u0 = v0 = w0 = wx = wy = wz = ax = ay = az = bx = by = hx = hy = 0, at y = 0, b.Clamped-free (CFCF):u0 = v0 = w0 = wx = wy = wz = ax = ay = az = bx = by = hx = hy = 0, at x = 0, and y = 0.u0 – v0 – w0 – wx – wy – wz – ax – ay – az – bx – by – hx – hy – 0, at x = a and y = b.Simply supported-clamped (SCSC):u0 = w0 = wy = ax = az = by = hx = 0, at x = 0 and y = 0.u0 = v0 = w0 = wx = wy = wz = ax = ay = az = bx = by = hx = hy = 0, at x = a, y = b.

5.1. Convergence and validation study

To make certain the accuracy and proficiency of the present finite element formulation, three test examples have beenanalyzed for free vibration of the FGM plates.

Fig. 16. Non-dimensional center deflection vs. volume fraction index n for simply supported and clamped (SSCC) Al/Al2O3 rectangular plate.

Fig. 15. Non-dimensional center deflection vs. volume fraction index n for simply supported (SSSS) Al/Al2O3 rectangular plate.


5.1.1. Free vibration analysis

Example 1. We first consider the accuracy of the present finite element formulation by comparing the results with thosegiven by Uymaz and Aydogdu [16] which is based on the small strain linear elasticity theory. A convergence study is alsopresented. A FGM plate with different boundary conditions is analyzed with the proposed finite element formulation. In thisexample, the analysis is performed with n = 5, aspect ratio a/b = 1 and thickness ratio a/h = 5, where n is the volume fractionindex and h is the thickness of the plate as defined earlier. The top face of the plate is ceramic-rich, whereas the bottom faceis metal rich. The plate is comprised of metal (aluminium) and ceramic (zirconia). The material properties as given in [16],are Eb = 70 � 109 N/m2, qb = 2702 kg/m3 for aluminium, and Et = 151 � 109 N/m2, qt = 3000 kg/m3 for zirconia. The Poisson’sratio m is assumed to be constant as 0.3. The non-dimensional frequency parameters are obtained by applying three-dimensional continuum method as: �x2 ¼ 12ð1� m2Þqcx2a2b2

=p4Ech2. Based on the convergence, it is concluded that (5 � 5)mesh is acceptable for free vibration analysis of the FGM plate. The accomplished results are in good agreement with thepublished results, as shown in Table 2.

Fig. 18. Non-dimensional deflection due to uniformly applied load vs. non-dimensional length for Al/ZrO2 rectangular FGM plate (a/h = 10) with simplysupported-clamped (SCSC) boundary condition.



Example 2. In this example, we consider the free vibration analysis of a square FGM plate made of zirconium oxide (ZrO2)and titanium alloy (Ti–6Al–4V), as given by Huang and Shen [30]. The top surface of the plate is assumed to be ceramic-rich,whereas the bottom surface is metal rich. The side and thickness of the square plate are a = 0.2 m and h = 0.025 m, respec-tively. Poisson’s ratio m is assumed to be constant and is equal to 0.3. Young’s modulus and mass density listed in Table 1. Theplate is simply supported at all edges and non-dimensional frequency parameter are defined as: �x ¼ xða2=hÞqmð1� m2Þ=Em� �1=2. The calculated frequency parameters are compared with Ref. [30], as shown in Table 3, again a goodagreement between the results are accomplished.

Example 3. In this example, we consider the FGM plate made of silicon nitride (Si3N4) and titanium alloy (SUS304), as givenby Zhao et al. [31] for free vibration. They employed first-order shear deformation plate theory to account for the transverseshear strain and rotary inertia. Mesh-free kernel particle functions are used to approximate the two-dimensional displace-ment fields. The side to thickness ratio (a/h) is equal to 10. The Poisson’s ratio for Si3N4 and SUS304 is assumed as 0.24 and0.3117, respectively. Young’s modulus and mass density are listed in Table 1. The plate is simply supported at all four edgesand non-dimensional frequency parameter are defined as: �x ¼ xa2=h qc=Ecð Þ1=2. The calculated frequency parameters arecompared with Ref. [31], as shown in Table 4, again a good agreement between the two results are obtained.

These three comparison studies show that the present result matches very well with the established one.




5.1.2. Static analysisTwo test examples have been analyzed for static analysis of the FGM plates to check the accuracy and proficiency of the

present finite element formulation.

Example 1. Here we consider the square FGM plate simply supported at all four edges with the proposed finite elementformulation. In this example, the analysis is performed for thickness ratio (a/h) = 5, and aspect ratio a/b = 1 with differentvolume fraction indices n = 0, 0.5, 1, 2 and1, where a and b are the sides of the plate and h is the thickness of the plate. Forthis problem, the convergence along with validation is also conducted. The various non-dimensionalized parameters usedare: central deflection, w/h, load parameter p0 ¼ p=Emh4 and thickness coordinate, �z ¼ z=h. The analysis is carried out forFGM plate made of aluminium and zirconia. The obtained results with the present outlined approach are presented in Table5 and compared with that of Ferreira et al. [18] that is based on the collocation multiquadric radial basis functions by a thirdorder shear deformation theory. It is observed that both results agree well. Based on the convergence, it is observed that(5 � 5) mesh is sufficient for static analysis. Hence for all computation, a (5 � 5) mesh is adopted.

Fig. 22. Non-dimensional deflection due to uniformly applied load vs. non-dimensional length for Al/ZrO2 rectangular FGM plate (a/h = 10) with clamped-free (CFCF) boundary condition.



Example 2. In this example, the numerical and analytical solutions for the centroid transverse deflection of the plate is com-pared with Qian et al. [32]. The top face of the plate is loaded by a normal pressure given by q0sinpx/asinpy/a. The plate iscomprised of Al/SiC with side to thickness ratio a/h = 5 and all edges are simply supported. The transverse displacement w

and thickness coordinate z have been non-dimensionalized as: �w ¼ 100Emh3

12a4ð1�m2mÞq0

w; �z ¼ 2zh . The non-dimensional center deflec-

tions with volume fraction index n along with the figure layout are preferred as used in Ref. [32] for direct comparison. Again,a good agreement is observed between the two results as shown in Fig. 4.

6. Results and discussion

Based on the analysis of foregoing sections it is observed that (5 � 5) mesh has been found to give good convergence forthe FGM plates as mentioned earlier. These have been used for accomplishing the results, unless otherwise stated. Tables 6and 7 show the variation of non-dimensional frequency parameter with the volume fraction index n for the Al/ZrO2 squareFGM plate. Choosing n = 0 and 1 corresponds to the frequency of the ceramics and metallic components, respectively. The




results are computed only for the first five modes with side to thickness ratios a/h = 10 for CCCC and SSSS boundary condi-tions. It is observed that as the volume fraction index increases the frequencies in all five modes decrease. This is due to thefact that, the larger volume fraction index means the plate has a smaller ceramic component and hence the stiffness is re-duced. It is also observed that increasing constraints on the boundaries increase the frequency parameter.

Variation of the frequency parameter with the volume fraction index of Al/ZrO2 FGM plate for a/h = 20 with CCCC andSSSS boundary condition is presented in Tables 8 and 9, respectively. It is noticed that frequency parameter decreases withthe increase of volume fraction index n as expected. It is anticipated that there is approximately 9% rise in frequency param-eter for CCCC boundary condition as a/h ratio changes from 10 to 20 and nearly 3% rise for SSSS boundary condition.

In Tables 10–13, the frequency characteristics for square SUS304/Si3N4 FGM plates are investigated for five modes only ata/h = 10 and 20 for CCCC and SSSS boundary conditions. It is observed that the frequency parameter decreases as volumefraction index n increases, as expected. This decrease is different for CCCC and SSSS boundary conditions considered. Theconsiderable decrease is found for CCCC boundary conditions. Table 14 shows the non-dimensional center deflection of CFCF,Al/ZrO2 FGM plates with volume fraction index n for aspect ratio (a/b) ranges from 0.5 to 3. The plates are subjected to asinusoidal load and side to thickness ratio (a/h) is equal to 5. It is found that the deflection increases as volume fraction indexn increases. This is due to the fact that the bending stiffness is the maximum for fully ceramic plate, i.e., (n = 0) and degradesgradually as n increases.

Fig. 5 illustrates the effect of volume fraction index and side to thickness ratio (a/h) on the fundamental natural frequencyparameter of Clamped square Al/ZrO2 FGM plates. It is observed that with a particular volume fraction index, the frequencyparameter increases as the side to thickness ratio increases. The rise in frequency parameter is observed up to around a/h = 20, beyond this no changes in frequency parameter are distinguished. Fig. 6 shows the variation of fundamental naturalfrequency parameter with a/h ratio and volume fraction index n for CCCC boundary conditions of rectangular Al/ZrO2 FGMplates. Bestowing to this figure the frequency parameter increases with increasing side to thickness (a/h) ratio, for (a/h) lessthan 20 as observed in Fig. 5.

Figs. 7 and 8 show the influence of volume fraction index and side to thickness ratio on the fundamental frequencyparameters for SSSS square and rectangular Al/Al2O3 plates, respectively, and influence of volume fraction index and sideto thickness ratio on the fundamental frequency parameters for SSCC square and rectangular Al/Al2O3 plates are shown inFigs. 9 and 10, respectively. It is found that the similar frequency parameter behavior is noticed to those obtained inFig. 5. It is therefore accomplished that effects of the side to thickness ratio on the natural frequency parameter of platesis independent of the variation in the volume fraction index.

Figs. 11–13 represent the variation of the non-dimensional central deflection with volume fraction index for the alumin-ium–alumina square plates having CCCC, SSSS and SSCC boundary conditions. The load parameter is taken as (+1). It can beascertain that all of the curves that represent the various boundary conditions show a similar behaviour with the deflectionenhances as the volume fraction index increases. This is due to the fact that the larger volume fraction index means the platehas a smaller ceramic component and hence the stiffness is reduced. Figs. 14–16 show the effects of volume fraction indexand side to thickness ratio on the non-dimensional central deflection for CCCC, SSSS and SSCC rectangular Al/Al2O3 plates. Itis observed that the rectangular plates have slightly higher central deflection than square plate for same boundary conditionsand material properties.

Figs. 17–20 show the non-dimensional deflection due to uniformly applied load vs. non-dimensional length for rectangu-lar plates (a/b = 0.5) with varying volume fraction index n. The FGM plate is made up of Al/ZrO2 and the side to thicknessratio (a/h) of the plate is varying from 5 to 20. Simply supported-clamped boundary conditions have been used in the anal-ysis. It is perceptible from the figure that the plate deflection lies between those made of ceramic and metal. The variation ofthe non-dimensional deflection due to uniformly applied load vs. non-dimensional length for rectangular plates (a/b = 0.5)with varying volume fraction index n is described in Figs. 21–24. The plate is comprised of Al/ZrO2 and the side to thicknessratio (a/h) of the plate is varying from 5 to 20 with CFCF boundary condition. The deflection characteristics shown in thesefour figures are similar to those in Figs. 17–20.

7. Conclusions

An extensive study of the free vibration and static analysis of square and rectangular functionally graded plates is pre-sented, which is based on the higher order shear deformation theory with a special modification in the transverse displace-ment in conjunction with finite element models. The systems of algebraic equations are derived using variational approachfor the free vibration and static problem. A C0 isoparametric Lagrangian element with 117 degrees of freedom per element isdeveloped and implemented for both the problems. Convergence and validation studies have been carried out to inculcatethe accuracy of the present formulation. The obtained result shows a good agreement with those available in the literatureranging from thin to thick plates. The following conclusions are noted from the typical results obtained for different volumefraction indices, the aspect ratios, the thickness ratios and different combinations of the boundary conditions.

� The frequency parameter increases with the increase in plates aspect ratio (a/b) and smaller side to thickness (a/h) ratio.� The frequency parameter decreases with the increases of volume fraction index, n.


� The frequency parameter increases with increasing (a/h) ratio for a/h 6 20 and becomes approximately insensitive to a/hratio greater than 40.� There is an increases in frequency parameter as support condition changes from all edges simply supported (SSSS) to all

edges clamped (CCCC).� The non-dimensional central deflection for a given thickness ratio increases as the volume fraction index n increases.� The non-dimensional central deflection decreases with the increase of plates aspect ratio (a/b).� The non-dimensional central deflection with non-dimensional length shows that the plates with intermediate properties

undergo corresponding intermediate values of center deflection.

References

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