Finite Elements in Analysis and Design 47 (2011) 394–401

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Finite Elements in Analysis and Design

0168-87

doi:10.1

� Corr

E-m

talha.ae

journal homepage: www.elsevier.com/locate/finel

Large amplitude free flexural vibration analysis of shear deformable FGMplates using nonlinear finite element method

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 28 August 2009

Received in revised form

26 July 2010

Accepted 17 November 2010Available online 6 January 2011

Keywords:

Functionally graded material

Higher order shear deformation theory

Finite element method

Large amplitude vibration

Green–Lagrange

4X/$ - see front matter & 2010 Elsevier B.V. A

016/j.finel.2010.11.006

esponding author. Tel.: +91 3222 283026; fa

ail addresses: rsmtalha@aero.iitkgp.ernet.in,

ro@yahoo.com (M. Talha).

a b s t r a c t

In this paper, large amplitude free flexural vibration analysis of shear deformable functionally graded

material (FGM) plates are investigated. The material properties of the FGM plates are assumed to vary

through the thickness of the plate by a simple power-law distribution in terms of the volume fractions of

the constituents. The nonlinear finite element equations are obtained using higher order shear

deformation theory with a special modification in the transverse displacement. The Green–Lagrange

nonlinear strain–displacement relation with all higher order nonlinear strain terms is included in the

formulation to account for the large deflection response of the plate. The fundamental equations are

obtained using variational approach by employing traction free boundary conditions on the top and

bottom faces of the plate. Results are obtained by employing an efficient C0 finite element with 13 degrees

of freedom (DOFs) per node. Convergence tests and comparison studies have been carried out to establish

the efficacy of the present model. The variation of nonlinear frequency ratio with the amplitude ratio

is highlighted for different thickness ratios, aspect ratios and volume fraction index with different

boundary conditions.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

The accomplishment of functionally graded material is therealization of contemporary and distinct functions that cannotbe achieved by the traditional composite materials. These areadvanced composite materials with a microscopically inhomoge-neous anatomy and are usually made from a mixtures of ceramicand metal using powder metallurgy techniques. Continuouschanges in their microstructure distinguish FGM from othertraditional composite materials. The material property of theFGM can be tailored to obtain the specific demand in differentengineering applications in order to exploit the advantage of theproperties of individual constituent. This is possible, because thematerial composition changes gradually in a preferred direction.The advantage of using this material is that it eliminates theinterface problem due to smooth and continuous change ofmaterial properties from one surface to other [1,2].

Large amplitude free flexural vibration (LAFFV) behavior of aplate arises in many engineering applications, particularly in thepanels of aircraft. When a structure is deflected substantially, i.e.,half of its thickness, a considerable geometrical nonlinearityoccurs, mostly due to the development of in-plane membrane

ll rights reserved.

x: +91 3222 255303.

stresses. These membrane stresses are tensile in nature that stiffensthe plate. This stiffening effect results in the rise of resonancefrequencies and change of mode shapes. Thus, the linear model isnot being capable to determine the behavior of the structurescompletely. Therefore, in the recent years geometrically nonlinearflexural vibration of plates have received considerable attentioncompared to static large deflection behavior of plates.

Since this area is fairly new, published literature on the nonlinearfree and forced vibrations of FGM plate is limited in number and mostof them are fascinated on linear problem. Reddy [3] presentedtheoretical formulation and finite element models (FEM) in the framework of third order shear deformation theory for static and dynamicanalyses of the FGM plates. Vel and Batra [4] presented a three-dimensional analytical solution of simply supported rectangular FGMplates for free and forced vibrations. Suitable displacement functionswhich satisfy boundary conditions are used to solve governingequations by employing the power series method. Efraim and Eisen-berger [5] obtained exact free vibration frequencies and modes ofvariable thickness thick annular FGM plates. Gunes and Reddy [6]investigated geometrically nonlinear analysis of circular FGM platessubjected to mechanical and thermal loads. They used Green–Lagrangestrain tensor with all its terms in the analysis.

Chen et al. [7] derived nonlinear partial differential equations for thevibration motion of initially stressed FGM plates. The formulation arederived for the nonlinear vibration motion of the FGM in a general stateof arbitrary initial stresses, based on classical laminated plate theory(CLPT). Talha and Singh [8] studied free vibration and static analysis of

a

b

x

y

z

Ceramic

Metal

+h/2

-h/2

Fig. 1. Geometry and dimensions of the plate.

M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401 395

FGM plates using modified HSDT kinematics. The fundamentalequations are obtained using variational principle by considering thestress free boundary conditions at the top and bottom faces of the plate.Yang and Shen [9] analyzed the free and forced vibration analyses forinitially stressed FGM plates in thermal environment. Temperaturedependent material properties are assumed for the analysis and theformulations are based on Reddy’s higher order shear deformationtheory, which includes the thermal effects due to uniform temperaturevariation. Sundararajan et al. [10] developed nonlinear formulationbased on von-Karman assumptions to study the free vibrationcharacteristics of FGM plates in thermal environment. They obtainednonlinear governing equations using Lagrange’s equations of motionand solved using FEM, coupled with direct iterative technique.

Chen [11] presented nonlinear vibration for FGM plate in thestate of non-uniform initial stresses. Galerkin method have beenused for transformation of the governing nonlinear partial differ-ential equations into ordinary nonlinear differential equations, andthe nonlinear and linear frequencies are obtained using the Runge–Kutta method.

Ng et al. [12] presented the parametric resonance of FGMrectangular plates under harmonic in-plane loading. It is foundthat the parametric resonance of FGM rectangular plate varies byvarying the power-law exponent, which controls the materialdistribution in the structures.

Allahverdizadeh et al. [13] developed a semi-analyticalapproach for nonlinear free and forced axi-symmetric vibrationsof a thin circular FGM plates. The formulation is based on the CLPTkinematics and the geometric nonlinearity is incorporated in von-Karman sense. Woo et al. [14] provided an analytical solution forthe nonlinear free vibration behavior of FGM plates. The governingequations for thin rectangular FGM plates are obtained using thevon-Karman theory for large transverse deflection, and mixedFourier series analysis is used to get the solution.

Huang and Shen [15] studied nonlinear vibration and dynamicresponse of FGM plates in thermal environments. The formulationsare based on the HSDT kinematics and general von-Karman typeequation, which includes thermal effects.

Yang and Shen [16] investigated large deflection and post-buckling responses of FGM rectangular plates by using semi-analy-tical approach under transverse and in-plane loads. The formulationsare based on the CLPT kinematics. Paraveen and Reddy [17]investigated the response of FGM plates using FEM that accountsfor the transverse shear strains, rotary inertia and moderately largerotations in the von-Karman sense. The gradation of properties isassumed according to power-law throughout the thickness andcomparisons have been made with homogeneous isotropic plates.

The determination of accurate nonlinear behavior of the FGMfundamentally depends on the theory used to model the structure.The classical laminated plate theory which requires that normals tothe mid-plane remain normal during plate deflections may beinappropriate for analysis of the FGM plates, in which volumefractions of two or more materials vary continuously as a functionof position in a preferred direction. The inaccuracy occurs due toneglecting the effects of transverse shear and normal strains in theplate [18]. Due to continuous variation in material properties, thefirst order shear deformation theory and higher order sheardeformation theory may be conveniently used in the analysis. Itis noted that the first order shear deformation theory proposed byMindlin [21] does not satisfy the parabolic variation of transverseshear strain in the thickness direction. Consequently, the solutionaccuracy of the FSDT depends on the shear correction factors.Therefore, it has to be incorporated to adjust the transverse shearstiffness. Generally, in the HSDT kinematics the in-plane displace-ments are assumed to be a cubic expression of the thicknesscoordinate and the out-of-plane displacement to be constant. In thepresent study, the structural model kinematics assumes the

cubically varying in-plane displacement over the entire thickness,while the transverse displacement varies quadratically to achievethe accountability of normal strain and its derivative in calculationof transverse shear strains. Thus, the development of higher ordershear deformation theory for describing the mechanical behavior ofFGM structures has been of high importance to the researchers. Inthis regard the geometrically nonlinear free flexural vibrations ofthe FGM plates using HSDT kinematics by incorporating geometricnonlinearity in Green–Lagrange sense is appropriate to examinethe responses of FGM structure accurately.

It is accomplished from the literature that the nonlinear freevibration analysis of the FGM plates using HSDT kinematics withgeometric nonlinearity in Green–Lagrange sense has not beenreported in the literature to the best of the authors’ knowledge. Thepresent research aims to develop a higher order shear deformationtheory with a special modification in transverse displacement thatprovides an additional freedom to the displacements through thethickness, and consequently eliminates the over prediction. Thegeometric nonlinearity in the formulation has been incorporatedby taking all higher order nonlinear strain terms associated withGreen–Lagrange theory. The governing equations are formulatedusing the variational approach. A nonlinear C0 continuous isopara-metric FEM is proposed to minimize the computational exerciserequired in the disposition of element matrices without compro-mising the solution credibility. The nonlinear fundamental fre-quencies are obtained for different thickness ratios, the aspectratios, the amplitude ratios, and for different volume fractionindices and boundary conditions. The obtained results are com-pared with those available in the literature.

2. Theoretical formulation

2.1. Geometric configuration and material properties

The FGM plate is regarded to be a single layer plate of uniformthickness. Here we ascertain the FGM plate of length a, width b andtotal thickness h made from an isotropic material of metal andceramics, in which the composition varies from top to bottomsurface as shown in Fig. 1. The top surface ðZ ¼ þh=2Þ of the plate isceramic rich, whereas the bottom surface ðZ ¼�h=2Þ is metal rich.The coordinates x, y are considered along the in-plane directionsand z along the thickness direction. The formulations are restrictedhere with the assumption of a linear elastic material behavior forsmall strains and large displacements.

The elastic material properties vary through the plate thicknessaccording to the volume fractions of the constituents. Power-lawdistribution is commonly used to describe the variation of material

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

M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401396

properties (Fig. 2), which is expressed as [19]

PðzÞ ¼ ðPc�PmÞVcþPm ð1Þ

VcðzÞ ¼z

hþ

1

2

� �n

ð0rnr1Þ 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 fractionexponent. The effective material properties of the plate including,Young’s modulus E, densityr vary according to Eq. (1) and Poisson’sratio n is assumed to be constant. The above power-law theoryestimates a simple rule of mixtures which is used to find theeffective properties of the ceramic–metal graded plate. This rule ofmixtures applies in the thickness direction only.

2.2. Displacement field and strains

In the present study, system of governing equations for FGM plate isderived by using variational approach. The origin of the materialcoordinates is at the middle of the plate as shown in Fig. 1. For accurateanalysis of transverse shear effects in the mathematical formulation theHSDT kinematics has been used, with a special modification in thetransverse displacement. The in-plane displacements u , v and thetransverse displacement w for the plate is assumed as

uðx,y,z,tÞ ¼ uðx,y,tÞþzcxðx,y,tÞþz2xxðx,y,tÞþz3rxðx,y,tÞ

vðx,y,z,tÞ ¼ vðx,y,tÞþzcyðx,y,tÞþz2xyðx,y,tÞþz3ryðx,y,tÞ

wðx,y,z,tÞ ¼wðx,y,tÞþzczðx,y,tÞþz2xzðx,y,tÞ ð3Þ

where t is the time, u, v, and w are the corresponding displacements of apoint on the mid-plane. cx and cy are the rotations of normal to themid-plane about the y-axis and x-axis, respectively. The functions xx,xy, rx and ry are the higher order terms in the Taylor series expansiondefined in the mid-plane of the plate. The higher order terms aredetermined by diminishing the transverse shear stresses txz ¼ t4 andtyz ¼ t5 on the top and bottom surfaces of the plate, and by applyingthis boundary condition the displacement field is modified as

u ¼ uþ f1ðzÞcxþ f2ðzÞaxþ f3ðzÞbxþ f4ðzÞyx

v ¼ vþ f1ðzÞcyþ f2ðzÞayþ f3ðzÞbyþ f4ðzÞyy

w ¼wþ f5ðzÞczþ f6ðzÞaz ð4Þ

whereax ¼ @cz=@x,bx ¼ @w=@x,yx ¼ @xz=@x,ay ¼ @cz=@y,by ¼ @w=@y,yy ¼ @xz=@y and xz ¼ az and f1ðzÞ ¼ C1z�C2z3, f2ðzÞ ¼ �C3z2,f3ðzÞ ¼�C4z3, f4ðzÞ ¼�C5z3, f5 ¼ C1z, f6 ¼ C1z2, C1 ¼ 1, C2 ¼ C4 ¼

4=3h2, C3 ¼ 1=2, C5 ¼ 1=3.The displacement vector for the model based on the modified

displacement field as given in Eq. (4) is written as

fLg ¼ ½u,v,w,cx,cy,cz,ax,ay,az,bx,by,yx,yy�T ð5Þ

2.3. Strain–displacement relation

The nonlinear Green–Lagrange strain–displacement relation forFGM plate can be represented as [20]

feg ¼

exx

eyy

ezz

gyz

gxz

gxy

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

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

¼

@u@x

@v@y

@w@z

@w@y þ

@v@z

@u@z þ

@w@x

@v@x þ

@u@y

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

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

þ1

2

@u@x

� �2þ @v

@x

� �2þ @w

@x

� �2� �

@u@y

� �2þ @v

@y

� �2þ @w

@y

� �2� �

@u@z

� �2þ @v

@z

� �2þ @w

@z

� �2� �

2 @u@y

� �@u@z

� �þ @v

@y

� �@v@z

� �þ @w

@y

� �@w@z

� �h i2 @u

@x

� �@u@z

� �þ @v

@x

� �@v@z

� �þ @w

@x

� �@w@z

� �h i2 @u

@x

� �@u@y

� �þ @v

@x

� �@v@y

� �þ @w

@x

� �@w@y

� �h i

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

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

ð6Þ

or, in compact form Eq. (6) may be written as

feg ¼ fe lgþfenlg ð7Þ

where fe lg and fenlg are the linear and nonlinear strain vectors,respectively. Substituting Eq. (6) in Eq. (3) the strain displacementof the FGM plate is expressed as

feg ¼

e01

e02

e03

e04

e05

e06

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

9>>>>>>>>>=>>>>>>>>>;þ

1

2

enl01

enl02

enl03

2enl04

2enl05

2enl06

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

9>>>>>>>>>>=>>>>>>>>>>;þz

k11

k12

k13

k14

k15

k16

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

9>>>>>>>>>=>>>>>>>>>;þ

1

2

knl11

knl12

knl13

2knl14

2knl15

2knl16

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

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

266666666664

377777777775

þz2

k21

k22

0

k24

k25

k26

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

9>>>>>>>>>=>>>>>>>>>;þ

1

2

knl21

knl22

knl23

2knl24

2knl25

2knl26

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

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

266666666664

377777777775

þz3

k31

k32

0

0

0

k36

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

9>>>>>>>>>=>>>>>>>>>;þ

1

2

knl31

knl32

knl33

2knl34

2knl35

2knl36

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

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

266666666664

377777777775þz4 1

2

knl41

knl42

knl43

2knl44

2knl45

2knl46

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

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

M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401 397

þz5 1

2

knl51

knl52

0

2knl54

2knl55

2knl56

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

9>>>>>>>>>>=>>>>>>>>>>;þz6 1

2

knl61

knl62

0

0

0

2knl66

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

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

ð8Þ

The above strain–displacement relation (Eq. (8)) can be expressedin terms of mid-plane strain vector by classifying the linear strainvector felg and nonlinear strains vector fenlg separately, as

feg ¼ felgþfenlg ¼ ½T�lfe lgþ12½T�nlfenlg ð9Þ

where

fe lg ¼ fe01 e0

2 e03 e0

4 e05 e0

6 k11 k1

2 k13 k1

4 k15 k1

6 k21 k2

2 k24 k2

5 k26 k3

1 k32 k3

6g

ð10Þ

fenlg ¼

�1

2

enl01 enl0

2 enl03 enl0

4 enl05 enl0

6 knl11 knl1

2 knl13 knl1

4 knl15 knl1

6 knl21 knl2

2 knl23 knl2

4

knl25 knl2

6 knl31 knl3

2 knl33 knl3

4 knl35 knl3

6 knl41 knl4

2 knl43 knl4

4 knl45 knl4

6 knl51 knl5

2

0 knl54 knl5

5 knl56 knl6

1 knl62 0 0 0 knl6

6

8>><>>:

9>>=>>;

ð11Þ

The linear strain vector in terms of mid-plane strain vector felg canbe further expressed as

felg20�1 ¼ ½w�20�13fLg13�1 ð12Þ

where ½w� and fLg are a matrix of differential operator anddisplacement field vector, respectively. The superscripts 0, 1, 2–3in Eq. (10) and nl0, nl1, and nl2–6 in Eq. (11) represents themembrane, bending and higher order terms, respectively.

2.4. Constitutive relations

The constitutive relation describes how the stresses and strainsare related within the plate and is expressed as

sxx

syy

szz

syz

sxz

sxy

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

9>>>>>>>>>=>>>>>>>>>;¼

Q11 Q12 Q13 0 0 0

Q12 Q22 Q23 0 0 0

Q13 Q23 Q33 0 0 0

0 0 0 Q44 0 0

0 0 0 0 Q55 0

0 0 0 0 0 Q66

26666666664

37777777775

exx

eyy

ezz

gyz

gxz

gxy

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

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

ð13Þ

where Q11¼Q22 ¼ Q33¼EðzÞð1�n2Þ=ð1�3n2�2n3Þ, Q12¼Q13¼Q23 ¼

EðzÞnð1þnÞ=ð1�3n2�2n3Þ, Q44 ¼Q55 ¼ Q66 ¼ EðzÞ=2ð1þnÞ. The elas-tic modulus E, and the coefficients Qij vary in the thickness directionof the plate according to Eqs. (1) and (2).

2.5. Strain energy of FGM plate

The strain energy of the FGM plate is given by

U ¼1

2

ZvfegTfsg dV ð14Þ

By substituting the strains and stresses from Eqs. (9) and (13), theabove equation becomes

U ¼1

2

ZVfelþenlg

Ti ½Q �felþenlgi dV ð15Þ

Eq. (15) in expanded form is written as

U ¼1

2

ZV

felgTi ½D1�felgiþ

12 felg

Ti ½D2�fenlgi

þ 12 fenlg

Ti ½D3�felgiþ

14 fenlg

Ti ½D4�fenlgi

!dA ð16Þ

where ½D1�¼R h=2�h=2½T�

Tl ½Q �½T�l dz, ½D2�¼

R h=2�h=2½T�

Tl ½Q �½T�nl dz, ½D3�¼

R h=2�h=2

½T�Tnl½Q �½T�l dz, ½D4� ¼R h=2�h=2½T�

Tnl½Q �½T�nl dz and [Q] is the stiffness

coefficients.

2.6. Kinetic energy

The kinetic energy of the FGM plate can be expressed as

T ¼ 12

ZVrf _u gTf _u gdV ð17Þ

where r and fug are the density and global displacement vector ofthe plate. The global displacement field model as given by Eq. (4)may be represented as

fug ¼ ½N �fLg ð18Þ

where fLg is as defined in Eq. (5), and the function of thickness co-ordinate ½N � may be represented as

½N � ¼

1 0 0 f1ðzÞ 0 0 f2ðzÞ 0 0 f3ðzÞ 0 f4ðzÞ 0

0 1 0 0 f1ðzÞ 0 0 f2ðzÞ 0 0 f3ðzÞ 0 f4ðzÞ

0 0 1 0 0 f5ðzÞ 0 0 f6ðzÞ 0 0 0 0

264

375 ð19Þ

Substituting Eq. (17) into Eq. (18), the kinetic energy becomes

T ¼ 12

ZA

ZZrf _LgT ½N �T ½N �f _Lg dz

� �dA¼ 1

2

ZAf _LgT ½m�f _Lg dA ð20Þ

3. Finite element implementation

A nine-noded quadrilateral C0 isoparametric element is used inthe present study. The domain is discretized into a set of finiteelements. Over each of the elements, the displacement vector andelement geometry of the model is expressed by

fLg ¼XNN

i ¼ 1

NifLgi; x¼XNN

i ¼ 1

Nixi; y¼XNN

i ¼ 1

Niyi ð21Þ

where Ni is the interpolation function (shape function) for the ithnode, fLgi is the vector of unknown displacements for the ith node,NN is the number of nodes per element and xi and yi are Cartesiancoordinate of the ith node.

3.1. Governing equation

The governing equation for nonlinear free vibration problem ofthe FGM plate can be derived using variational principle, which isthe generalization of the principle of virtual displacement.Lagrange equation for a conservative system can be written as

d

dt

@T

@f _qig

� �þ

@U

@fqigþ

@

@fqig

g2

Z ax�@cz

@x

� �2þ bx�

@cz

@x

� �2

þ yx�@xz

@x

� �2þ ay�

@cz

@y

� �2

þ by�@w@y

� �2þ yy�

@xz

@y

� �2

26666664

37777775

dv

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

9>>>>>>=>>>>>>;¼ 0

ð22Þ

for i¼ 1,2, . . . where T is the kinetic energy, U is the strain energy,fqig and f _qig are the generalized coordinates and generalizedvelocities, respectively. The equilibrium equation for nonlinearfree vibration analysis with large deformation (i.e., small strainsand large displacements relation) can be represented as

½M�f €qgþð½KþgKc�þ12 ½Knl1�þ½Knl2�þ

12½Knl3�Þfqg ¼ 0 ð23Þ

where ½M�,g,½Kc�,½K�,½Knl1�,½Knl2�,½Knl3�, and fqg, are the global massmatrix, penalty parameter that enforces constraints, the global

M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401398

linear stiffness matrix arising due to constrains, the global linearstiffness matrix, the global nonlinear stiffness matrices and globaldisplacement vector, respectively.

Assuming the system vibrating in principal mode with naturalfrequency o, Eq. (23) can be conveniently reduced to nonlineargeneralized eigenvalue problem as

½K�fqg ¼ l½M�fqg ð24Þ

with l¼o2, where o is defined as frequency of natural vibration.This eigenvalue problem is solved by employing direct iterativeprocedure. Initially, the linear response with normalized first modeis calculated by solving Eq. (24) with all nonlinear [Knl1], [Knl2], and[Knl3] terms fixed to zero. This normalized vector is scaled up insuch a way that the maximum deflection is equal to the desiredamplitude w/h (w is the maximum transverse displacement and h isthe thickness of the plate). After that nonlinear stiffness matrices[Knl1], [Knl2], and [Knl3] are obtained with the first mode. Then usingupdated stiffness matrices, eigenvalue and its eigenvector areobtained. This attained eigenvector is used to compute the non-linear stiffness matrices for the next iteration. This exercisecontinues till two eigenvalues from the two subsequent iterationsreached the tolerance limit r10�3. The convergence is supposed toreach at said tolerance limit, and the corresponding o is thenonlinear frequency (onl) of the vibrated FGM plate.

Table 2Comparison of fundamental nonlinear frequency ratio ðonl=olÞ for the various

values of amplitude ratios of a square SSSS (Ti–6AL–4V/ZrO2) FGM plate with

different mess sizes.

Mesh size Amplitude ratio ðwmax=hÞ

0.2 0.4 0.6 0.8 1.0

Present (2�2) 1.0493 1.1597 1.2951 1.4864 1.6333

Present (3�3) 1.0476 1.1468 1.2857 1.4492 1.6094

Present (4�4) 1.0467 1.1405 1.2786 1.4300 1.6064

Present (5�5) 1.0455 1.1409 1.2765 1.4244 1.6055

Huang and Shen [15] 1.021 1.082 1.176 1.296 1.436

4. Numerical results

In the present work, large amplitude free flexural vibrationbehavior of the FGM plates has been addressed. The LAFFVresponses are computed using the proposed mathematical modelin conjunction with nonlinear FEM. The geometric nonlinearity isincorporated in the formulation applying Green–Lagrange theoryby taking all higher order terms. A computer program has beendeveloped in MATLAB 7.5.0 (R2007b) environment. The validationand efficacy of the proposed algorithm is examined by comparingthe results with those available in the literature. A nine-nodedLagrange isoparametric element, with 117 degrees of freedom perelement for the present HSDT kinematics has been used fordiscretizing the plate. Full integration schemes (3�3) and selectiveintegration schemes (2�2) are used for thick and thin plates,respectively, to compute the results.

In the present analysis simply supported boundary conditionsare used to check the efficacy of the model. However, the formula-tion and code does not impose any limitations. The properties of theFGM constituents at room temperature (300 K) are shown inTable 1, which have been used for the computation of the resultsthroughout the study, unless specified otherwise. The features ofvolume fraction of the ceramic phase through the dimensionlessthickness is depicted in Fig. 2. In the analysis, it is assumed that thematerials are perfectly elastic throughout the deformation.

Table 1Properties of the FGM components.

Material Properties

E (N/m2) n r ðkg=m3Þ

Aluminium (Al) 70�109 0.30 2707

Alumina (Al2O3) 380�109 0.30 3800

Zirconia (ZrO2) 151�109 0.30 3000

Silicon nitride (Si3N4) 427�109 0.28 3210

Titanium alloy (Ti–6Al–4V) 105.7�109 0.298 4429

Stainless steel(SUS304) 207.78�109 0.28 8166

The following sets of boundary conditions are considered in thepresent study:

Simply supported: (SSSS)

u0 ¼w0 ¼cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0 at x¼ 0 and a

v0 ¼w0 ¼cx ¼ ay ¼ az ¼ bx ¼ yy ¼ 0 at y¼ 0 and b

Clamped: (CCCC)

u0 ¼ v0 ¼w0 ¼cx ¼cy ¼cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0

at x¼ 0,a and y¼ 0,b

Simply supported and clamped: (SCSC)

u0 ¼w0 ¼cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0 at x¼ 0 and y¼ 0

u0 ¼ v0 ¼w0 ¼cx ¼cy ¼cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0

at x¼ a, y¼ b

Simply supported and clamped: (SSCC)

u0 ¼w0 ¼cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0 at x¼ 0 and a

u0 ¼ v0 ¼w0 ¼cx ¼cy ¼cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0

at y¼ 0,b

4.1. Convergence and validation study

The accuracy of the present finite element formulation isvalidated by comparing the results with those available in theliterature [15] which is based on the HSDT and general von-Karmantype of nonlinearity. A convergence study is also presented. A FGMsquare plate, simply supported at all four edges is analyzed. In thisexample, the analysis is performed with volume fraction indexn¼2, aspect ratio a=b¼ 1 with sides of the plate a¼ b¼ 0:2 m, fordifferent amplitude ratios (wmax=h). wmax is the maximum deflec-tion at the center of the plate. The FGM plate comprised titaniumalloy, Ti–6Al–4V (metal) and zirconia, ZrO2 (ceramic). The resultshave been carried out in terms of frequency ratio onl=ol withvarious mesh divisions as shown in Table 2. This clearly shows that

% Difference 2.343 5.162 7.873 9.014 10.557

Table 3Comparison of fundamental nonlinear frequency ratio ðonl=olÞ for the various

values of amplitude ratios of a square SSSS (Si3N4/SUS304) FGM plate (a/h¼10).

ðwmax=hÞ FEM

Ref. [10] Present % Difference

0.2 1.0063 1.0198 1.323

0.4 1.0654 1.1164 3.712

0.6 1.1707 1.2377 5.413

0.8 1.3155 1.4214 7.452

1.0 1.4789 1.6390 9.768

M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401 399

the solution accuracy and the rate of convergence with meshrefinement are good for frequency ratio onl=ol for the differentvalues of amplitude ratios (wmax=h). Based on the convergence, it isconcluded that ð5� 5Þmesh is sufficient for LAFFV analysis. Table 3shows the comparison of the frequency ratio onl=ol for Si3N4/SUS304 simply supported square FGM plate. The side to thicknessratio (a/h) is taken as 10. The top surface of the plate is ceramic rich(Si3N4), and bottom surface is metal rich (SUS304). It can beobserved that the solutions from this study agrees well with thosepresented by Sundararajan et al. [10]. It is accomplished fromTables 2 and 3, as the amplitude ratio (wmax=h) increases, the %difference in frequency ratioonl=ol increases. This is due to the factthat Refs. [15,10], had incorporated the geometric nonlinearity invon-Karman sense. Whereas, in the present study geometricnonlinearity is included in Green–Lagrange sense with all higherorder nonlinear strain terms. However, it has been observed fromthe detailed analysis that, the von-Karman type of geometricnonlinearity gives fairly good results up to ðwmax=hÞ ¼ 0:6, for thesakeof conciseness it is not shown here. The difference in the resultsuggests the requirement of a better mathematical model for thestructures which undergo small strain having large deformationand/or large rotation.

Table 5Effect of variation of thickness ratios (a/h ¼ 5–100) with the volume fraction index n

for (Ti–6AL–4V/ZrO2) FGM plates consisting with SSSS boundary condition.

a/b n a/h Amplitude ratio ðwmax=hÞ

0.5 1.0 1.2 1.4 1.6 1.8 2.0

1 0 5 2.1878 2.3648 2.5774 2.5253 2.8965 2.9843 2.9779

15 1.5223 1.9828 2.5139 2.0053 2.1878 2.3212 2.4036

25 1.1636 1.5159 1.7680 1.9931 2.0526 2.2484 2.2014

50 1.1662 1.5809 1.7383 1.8765 1.9418 2.0165 2.1394

100 1.5491 1.5291 1.7330 1.7827 1.8225 2.1086 1.8534

1 1 5 1.9459 2.3176 2.4829 2.3710 2.7433 2.5920 2.7165

15 1.4192 1.6088 1.7816 1.8540 2.1428 2.2791 2.3801

25 1.3077 1.6066 1.7874 1.7932 1.8241 2.1910 2.1139

50 1.2114 1.5622 1.7250 1.8279 1.5265 2.1771 2.1221

100 1.2080 1.4951 1.6407 1.7625 1.8041 1.8915 1.9076

5. Parametric study

Based on the established approach and analysis of aforemen-tioned sections, the simply supported (SSSS), clamped (CCCC),simply supported–simply supported and clamped–clamped (SSCC)and simply supported and clamped (SCSC) boundary conditionshave been employed in the analysis. It is also observed that ð5� 5Þmesh gives good convergence as mentioned earlier, and have beenused for accomplishing the results, unless otherwise stated.

Table 4 shows the effect of volume fraction index n and thicknessratio a=h¼ 10 on the frequency ratio (onl=ol) of a SSSS square titaniumalloy (Ti–6AL–4V) and zirconium oxide (ZrO2) FGM plate. The proper-ties of the constituents are provided in Table 1 as mentioned earlier. Thetop surface of the plate is ceramic rich whereas the bottom surface ismetal rich. The non-dimensional linear frequency is assumed aso ¼oða2=hÞ½rmð1�n2Þ=Em�

1=2. It is observed from the table that thefrequency ratio decreases with the increase in the volume fractionindex n. This decrease in frequency ratio is on expected lines, becauseincrease in the volume fraction index n means that a plate has a smallerceramic component, and thus its stiffness is reduced and consequentlyits non-dimensional fundamental linear frequency is decayed. It can beadditionally viewed that the frequency ratio enhances with the rise inamplitude ratio.

Table 5 shows the effect of variation of thickness ratios(a=h¼ 52100Þ, ranging from thick to thin plates, with the volumefraction index n. Here, FGM plates consisting of (Ti–6AL–4V/ZrO2) withall simply supported edges. The aspect ratio is taken as a=b¼ 1 withvolume fraction index n¼ 0 and 1. Choosing n index as zero

Table 4

Effect of volume fraction index n and thickness ratio ða=h¼ 10Þ on the fundamental no

n Amplitude ratio ðwmax=hÞ

0.5 1.0 1.2 1.4

0 1.2144 1.6953 1.7564 1.9675

0.2 1.1989 1.6266 1.7070 1.8837

0.4 1.1613 1.5997 1.6052 1.8285

0.6 1.1406 1.5757 1.6156 1.7887

0.8 1.0626 1.5438 1.8075 1.7566

1.0 1.1250 1.5328 1.8160 1.9892

corresponds to the fully ceramic plate and 1 complies linear variationof ceramic and metal constituent throughout the thickness of the plate.It is observed that the frequency ratio (onl=ol) decreases with theincrease of thickness ratio and it shows more significant differences atlower thickness ratio than larger thickness ratio. This implies that thehigher order theory is more suitable for thick plates and higher orderterms must be considered for the thick plate condition. It is alsorevealed that the nonlinear frequency ratio is slightly higher for fullyceramic plate i.e., (n¼ 0) considered here, than (n¼ 1). It is alsoaccomplished that the frequency ratio increases with the increase ofamplitude ratio. However, mixed types of behavior on the frequencyratio have also been observed due to the presence of severe non-linearity. Therefore, it can be concluded that there is a great need toanalyze the structures with geometric nonlinearity in Green–Lagrangesense to closely monitor the response of structures.

Table 6 shows the variation of frequency ratio (onl=ol) withamplitude ratio (wmax=h) for SCSC square stainless steel (SUS304)and silicon nitride (Si3N4) FGM plate. The fundamental linearfrequency is non-dimensionalized as o ¼oða2=hÞðrc=EcÞ

1=2. It isobserved from the table that the frequency ratio decreases with theincrease in the volume fraction index n up to a certain value, sayn¼ 2, and then the frequency ratio (onl=ol) increases with addi-tional increase in volume fraction index n. It is realized that thevalue of linear frequency decreases for higher value of index n andaccordingly the nonlinear frequency decreases. It is also observedthat the degradation in linear frequency at higher volume fractionindex n is more than that of the nonlinear frequency. Hence, theoverall trend of frequency ratio (onl=ol) increases. The frequencyratio is on higher side for a=h¼ 10 compared to a=h¼ 20.

Table 7 describes the frequency ratio (onl=ol) for aluminum (Al)zirconia (ZrO2) square FGM plate for different volume fractionindex n and thickness ratios a=h¼ 10,20. Here it is ascertained thatthe SSCC boundary conditions and the fundamental linear fre-quency are non-dimensionalized as o ¼oða2=hÞðrc=EcÞ

1=2. The

nlinear frequency ratio ðonl=olÞ of a square SSSS (Ti–6AL–4V/ZrO2) FGM plate.

1.6 1.8 2.0 2.5 3.0

2.0297 2.1317 2.2563 2.3442 3.2223

1.8896 1.9528 2.1609 2.2769 2.4834

1.7956 1.7913 2.1533 2.0778 2.2024

1.7815 1.7658 2.0857 1.9838 2.2071

1.8318 1.8659 2.0083 1.7678 2.1853

1.7559 1.8536 1.9366 1.8902 2.0427

Table 6

Effect of volume fraction index n and thickness ratio (a=h¼ 10,20) on the

fundamental nonlinear frequency ratio ðonl=olÞ of a square SCSC (SUS304/Si3N4)

FGM plate.

a/h n Amplitude ratio ðwmax=hÞ

0.4 0.8 1.2 1.6 2.0 2.5 3.0

10 0 1.8021 2.3109 2.6300 2.8015 3.0557 3.4309 3.7179

0.5 1.5874 1.0008 1.5458 1.6314 2.9435 3.13089 3.5417

1.0 1.4142 1.2675 1.5129 1.6463 1.8700 2.0262 2.1559

2.0 1.0170 1.2644 1.5040 1.6929 1.7843 2.1517 2.2543

5.0 1.3258 1.2858 1.4767 1.5057 1.8317 2.1517 2.2543

10 1.6972 1.4295 1.5295 1.6933 1.9354 2.1743 3.3274

20 1.7078 2.1414 1.5363 1.7776 2.0201 2.3155 3.4180

20 0 1.1703 1.2342 1.4923 1.7982 1.9464 2.1040 2.6612

0.5 1.1439 1.1953 1.4240 1.7515 1.8907 1.9308 2.0971

1.0 1.1851 1.2575 1.4339 1.8003 1.8464 1.9821 1.9718

2.0 1.0328 1.2820 1.3666 1.7797 1.7966 1.6951 1.8074

5.0 1.3726 1.2919 1.4855 1.4880 1.8993 1.9368 2.0235

10 1.3783 1.2919 1.4841 1.6734 1.9651 2.0987 2.0094

20 1.6682 1.4623 1.5517 1.7034 2.0167 2.1705 2.3860

Table 7

Effect of volume fraction index n and thickness ratio ða=h¼ 10,20Þ on the funda-

mental nonlinear frequency ratio ðonl=olÞ of a square SSCC (Al/ZrO2) FGM plate.

a/h n Amplitude ratio ðwmax=hÞ

0.4 0.8 1.2 1.6 2.0 2.5 3.0

10 0 1.2879 1.6377 1.8887 1.6682 1.7069 1.8243 1.9261

0.5 1.2263 1.2516 1.4220 1.6014 1.6600 1.6789 1.8934

1.0 1.0790 1.2415 1.4197 1.6472 1.5719 1.4982 1.7899

2.0 1.0697 1.2372 1.4792 1.5358 1.5892 1.6344 1.7471

5.0 1.6017 1.3289 1.5387 1.6404 1.7025 1.8584 1.9465

10 1.7078 1.5407 1.7285 1.7400 1.8018 2.0114 2.0812

20 0 1.0567 1.0921 1.4404 1.6349 1.8816 1.8294 1.9375

0.5 1.1672 1.2404 1.4658 1.6680 1.7973 1.7620 2.1963

1.0 1.0625 1.2209 1.3774 1.6434 1.7420 1.6145 2.0012

2.0 1.0711 1.2118 1.3295 1.5458 1.7123 1.6254 1.7120

5.0 1.0820 1.2751 1.4282 1.5680 1.7458 1.7045 1.7294

10 1.1137 1.3337 1.4963 1.7200 1.8116 1.7907 1.8068

Table 8

Effect of volume fraction index n and thickness ratio ða=h¼ 10,20Þ on the funda-

mental nonlinear frequency ratio ðonl=olÞ of a square CCCC (Al/ZrO2) FGM plate.

a/h n Amplitude ratio ðwmax=hÞ

0.4 0.8 1.2 1.6 2.0 2.5 3.0

10 0 1.9684 2.0164 2.1732 2.2922 2.3682 2.4913 2.6415

0.5 1.6091 2.0060 1.8838 2.1279 1.8076 2.0541 2.3469

1.0 1.4210 1.5484 1.3118 2.0159 1.6685 2.0171 2.3221

2.0 1.1243 1.3296 1.2678 1.6241 1.6463 1.9837 2.1529

5.0 1.2078 1.6388 1.3404 1.3452 1.3237 1.5808 1.6261

10 1.0433 1.7939 1.3167 1.7074 1.4038 1.7607 1.8508

20 0 1.3890 1.4091 1.3681 1.4796 1.7224 1.9074 1.9950

0.5 1.2011 1.3644 1.3396 1.5293 1.7078 1.8938 1.9192

1.0 1.1774 1.2441 1.3141 1.4864 1.5947 1.6689 1.7569

2.0 1.0718 1.1538 1.2878 1.2739 1.3288 1.5686 1.6802

5.0 1.1739 1.1704 1.3160 1.4655 1.5160 1.7768 1.7035

10 1.2796 1.2395 1.4699 1.4930 1.6376 1.7981 2.0432

M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401400

value of volume fraction index n varies from 0 to 10. It is observedthat the frequency ratio increases with the rise of amplitude ratio(wmax=h). However, it is ascertained that there is sudden drop in the

increasing frequency trend at a particular amplitude ratio and thensteadily increases with further increase in amplitude ratio showinghardening type of character. This behavior is most likely due to thechange in stiffness values, and apparently the redistribution ofmode shapes at certain level of amplitude of vibrations. It is alsorevealed that frequency ratio (onl=ol) first decreases with theincrease of volume fraction index n and again increases with theincrease of index value n, say ðn¼ 10Þ.

Table 8 shows the effect of volume fraction index n with thefrequency ratio (onl=ol) for (Al/ZrO2) square FGM plate fordifferent volume fraction index n and thickness ratiosa=h¼ 10,20. The plate is clamped at all its edges. It is seen thatthe frequency ratio decreases with the increase of thickness ratio(a/h). A mixed type of behavior in the frequency ratio (onl=ol) isobserved.

6. Conclusions

The nonlinear free flexural vibration analysis of the FGM plate isanalyzed using higher order shear deformation theory with a specialamendment in the transverse displacement associated with non-linear FEM in the capacity of Green–Lagrange theory. To implementthis nonlinear model a nine-noded C0 continuous isoparametricLagrangian element with 13 DOFs per node is developed andapplied to accomplish the frequency ratio by utilizing directiterative method. The governing equations are derived using thevariational approach. Convergence and validation studies have beencarried out to ascertain the accuracy of the present formulation.Numerical results for different aspect ratios, the thickness ratios, thevolume fraction indices and different combinations of the boundaryconditions have been presented. The results show the necessity andimportance of the higher order nonlinear terms.

Acknowledgments

The authors gratefully acknowledge the financial support bythe All India Council for Technical Education (AICTE), New Delhi(F. no. 1-10/RID/NDF-PG(19)/2008-09, dated 13th March 2009) astatutory body of the Government of India.

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