linear free flexural vibration of cracked functionally graded plates in thermal environment

Computers and Structures 89 (2011) 1535–1546

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Computers and Structures

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Linear free flexural vibration of cracked functionally graded platesin thermal environment

S. Natarajan a, P.M. Baiz b, M. Ganapathi c, P. Kerfriden a, S. Bordas a,⇑a Institute of Mechanics and Advanced Materials, Theoretical and Computational Mechanics, Cardiff University, Wales, UKb Department of Aeronautics, Imperial College, London, UKc Functional Head – Stress & DTA, Bombardier India Centre, IES – Aerospace Engineering, Mahindra Satyam, Bangalore, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 July 2010Accepted 10 April 2011Available online 17 May 2011

Keywords:Functionally graded plateLinear vibrationCracked plateRessiner–Mindlin plateShear flexible elementGradient index

0045-7949/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruc.2011.04.002

⇑ Corresponding author. Address: Institute of MechaTheoretical and Computational Mechanics, Cardiff ScBuilding, Room S1.03, CF24 3AA Wales, UK. Tel.: +44

E-mail address: [email protected]: http://www.engin.cf.ac.uk/whoswho/profil

www.researcherid.com/rid/A-1858-2009 (S. Bordas).

In this paper, the linear free flexural vibrations of functionally graded material plates with a throughcenter crack is studied using an 8-noded shear flexible element. The material properties are assumedto be temperature dependent and graded in the thickness direction. The effective material propertiesare estimated using the Mori–Tanaka homogenization scheme. The formulation is developed based onfirst-order shear deformation theory. The shear correction factors are evaluated employing the energyequivalence principle. The variation of the plates natural frequency is studied considering variousparameters such as the crack length, plate aspect ratio, skew angle, temperature, thickness and boundaryconditions. The results obtained here reveal that the natural frequency of the plate decreases withincrease in temperature gradient, crack length and gradient index.

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1. Introduction

A functionally graded material (FGM) is a new class of materialwhose properties are characterized by the volume fraction of itsconstituent materials. The concept of characterization of materialproperties is not new, but the unique feature of FGMs is that thesematerials are made of a mixture of ceramics and metals that arecharacterized by the smooth and continuous variation in propertiesfrom one surface to another [1–3]. For structural integrity, FGMsare preferred over fibre–matrix composites that may result in deb-onding due to the mismatch in the mechanical properties acrossthe interface of two discrete materials bonded together. With theincreased use of these materials for structural components in manyengineering applications, it is necessary to understand the dynamiccharacteristics of functionally graded plates.

Many researchers have recently attempted to study the bendingbehavior of FGM plates on three-dimensional elasticity solutions[4–7]. All these works are limited to simply supported plates undersinusoidal transverse mechanical or thermal loading. Reddy andCheng [4] and Vel and Batra [5] have accounted for the variation

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of material properties through the thickness according to apower-law distribution and the locally effective material proper-ties were obtained in terms of the volume fractions of the constit-uents through the Mori–Tanaka homogenization scheme.Kashtalyan [6] derived the elasticity solutions making use of thePlevako general solution of the equilibrium equations for inhomo-geneous isotropic media, whereas, Pan [7] studied the laminatedfunctionally graded simply supported rectangular plates undersinusoidal surface load, extending the Pagano’s solutions whichmay not be valid for finding the solutions of such plate problemswith continuous inhomogeneity. Elishakoff and Gentilini [8] inves-tigated the three-dimensional static analysis of clamped function-ally graded plates under uniformly distributed load applying theRitz energy method.

Application of 3D analysis, in general, is quite cumbersomewhile dealing with complex loading and boundary conditions.Hence, the analysis of isotropic, composite and FGM plates is car-ried out numerically as well as analytically using plate theoriesassuming plane stress conditions. Such approximation can predictglobal displacement and bending moments with sufficientaccuracy [9,10]. Few analytical and finite element studies on thebending analysis of FGM plates are recently available in the litera-ture using plate theories. Qian et al. [11] and Matsunaga [12]examined the bending of thick square FGM plates consideringthe higher-order shear deformation theory, whereas, Zenkour[13,14] dealt with 2D trigonometric functions based shear defor-mation theory. The nonlinear thermo-mechanical response of

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1536 S. Natarajan et al. / Computers and Structures 89 (2011) 1535–1546

FGM plates was examined by Praveen and Reddy [15] and Reddy[16] considering the higher-order structural theory. Carrera et al.[17] have obtained closed form and finite element solutions forthe static analysis of functionally graded plates subjected to trans-verse mechanical loads. The unified formulation employed in [17]permits a large variety of plate models with variable kinematicassumptions covering first-order as well as higher-order theories.However, higher-order models that involve additional displace-ment fields may be based on either an equivalent single layertheory or discrete layer approach. Furthermore, they are computa-tionally expensive in the sense that the number of unknowns to besolved is high compared to that of the first-order shear deforma-tion formulation.

Fig. 1. (a) Coordinate system of a rectangular FGM plate, (b) coordinate system of askew plate.

Fig. 2. Plate with a

In literature, there has been many studies on dynamic charac-teristics of FGM plates [15,18–24] and shells [25–29]. The dynamiccharacteristics of a cracked structural element is especially impor-tant because a crack in a vibrating structure results in stiffness de-crease, stress concentration, anisotropy and local flexibility, whichare functions of the crack location and size. Moreover the crack willopen and close depending on the vibration amplitude. The vibra-tion of cracked isotropic plates was studied as early as 1969 byLynn and Kumbasar [30] who used a Green’s function approach.Later, in 1972, Stahl and Keer [31] studied the vibration of crackedrectangular plates using elasticity methods. The other numericalmethods that are used to study the dynamic response and instabil-ity of plates with cracks or local defects are: (1) finite fourier seriestransform [32]; (2) Rayleigh–Ritz method [33]; (3) harmonic bal-ance method [34]; (4) finite element method [35,36]; (5) extendedfinite element method [37]; (6) smoothed finite element methods[38–42] and (7) Meshfree methods [43,11,44,45]. Recently, Yangand Chen [46] and Kitipornchai et al. [47] studied the dynamiccharacteristics of FGM beams with an edge crack. In case of FGM,the dynamic characterisitcs also depend on the gradient indexcompared to the isotropic case. However, to the author’s knowl-edge, studies of cracked FGM plates are scarce in the literature thatare of practical importance to the designers.

The main focus of this paper is to compute the linear free flex-ural vibrations of FGM plates with a through center crack using thefinite element method. Here, an eight-noded shear flexible quadri-lateral plate element based on field consistency approach [48,49] isused to study the dynamic characteristics of FGM plates subjectedto thermo-mechanical loadings. The shear correction factor is cal-culated from the energy equivalence principle. The temperaturefield is assumed to be constant in the plate and varied only inthe thickness direction. The material is assumed to be temperaturedependent and graded in the thickness direction according to apower law distribution in terms of the volume fractions of the con-stituents. The effective material properties are estimated from thevolume fractions and the material properties of the constituentsusing the Mori–Tanaka homogenization method [50,51]. The for-mulation developed herein is validated considering different prob-lems for which the solutions are available in the literature. Detailedparametric studies are carried out to understand the free vibrationcharacteristics of cracked FGM plates.

The paper is organized as follows. In Section 2, a brief introduc-tion to FGM is given followed by the plate formulation. The treat-ment of boundary conditions for skew plates and eight-nodedquadrilateral plate element are discussed in Section 3. A detailednumerical study is presented in Section 4 followed by conclusionin the last section.

center crack.

S. Natarajan et al. / Computers and Structures 89 (2011) 1535–1546 1537

2. Theoretical development and formulation

2.1. Functionally graded material

A functionally graded material (FGM) rectangular plate (lengtha, width b and thickness h), made by mixing two distinct materialphases: a metal and ceramic is considered with coordinates x, yalong the in-plane directions and z along the thickness direction(see Fig. 1). The material on the top surface (z = h/2) of the plateis ceramic and is graded to metal at the bottom surface of theplate (z = �h/2) by a power law distribution. The homogenizedmaterial properties are computed using the Mori–Tanaka scheme[50,51].

2.1.1. Estimation of mechanical and thermal propertiesBased on the Mori–Tanaka homogenization method, the effec-

tive bulk modulus K and shear modulus G of the FGM are evaluatedas [50–52,11]:

K � Km

Kc � Km¼ Vc

1þ 1� Vcð Þ 3 Kc�Kmð Þ3Kmþ4Gm

;

G� Gm

Gc � Gm¼ Vc

1þ 1� Vcð Þ Gc�Gmð ÞGmþf1

;

ð1Þ

Table 1Details of parameters used in the numerical study.

Parameter Assumed values

Gradient index, n 0, 0.5, 1,2, 5, 10Ceramic temperature (K), Tc 300, 400, 600Metal temperature (K), Tm 300, 300, 300Crack length c/a 0.2, 0.4, 0.6Aspect ratio a/b 1, 2Thickness of the plate a/h 10, 20Skew angle of the plate w (in degree) 0�, 15�, 30�, 45�Boundary conditions Simply supported and clamped

0

0.2

0.4

0.6

0.8

1

-0.4 -0.2

V c

gradient index, nn=0.5n=1n=2n=5n=10

Fig. 3. Through thickness vari

where

f1 ¼Gm 9Km þ 8Gmð Þ

6 Km þ 2Gmð Þ : ð2Þ

Here, Vi(i = c,m) is the volume fraction of the phase material. Thesubscripts c and m refer to the ceramic and metal phases, respec-tively. The volume fractions of the ceramic and metal phases are re-lated by Vc + Vm = 1, and Vc is expressed as:

VcðzÞ ¼2zþ h

2h

� �n

; n P 0; ð3Þ

where n in Eq. (3) is the volume fraction exponent, also referred toas the gradient index. The effective Young’s modulus E and Poisson’sratio m can be computed from the following expressions:

E ¼ 9KG3K þ G

;

m ¼ 3K � 2G2 3K þ Gð Þ :

ð4Þ

The effective heat conductivity coefficient j and coefficient of ther-mal expansion a is given by [53,54]:

j� jm

jc � jm¼ Vc

1þ 1� Vcð Þ jc�jmð Þ3jm

;

a� am

ac � am¼

1K � 1

Km

� �1

Kc� 1

Km

� � :ð5Þ

The effective mass density q is given by the rule of mixtures as [20]:

q ¼ qcVc þ qmVm: ð6Þ

2.1.2. Temperature distribution through the thicknessThe material properties P that are temperature dependent can

be written as [55]:

P ¼ Po P�1T�1 þ 1þ P1T þ P2T2 þ P3T3� �

; ð7Þ

0 0.2 0.4

z/h

ation of volume fraction.


where Po, P�1, P1, P2, P3 are the coefficients of temperature T and areunique to each constituent material phase. The temperature varia-tion is assumed to occur in the thickness direction only and thetemperature field is considered to be constant in the xy-plane. Insuch a case, the temperature distribution along the thickness canbe obtained by solving a steady state heat transfer equation:

� ddz

jðzÞdTdz

� �¼ 0; T ¼ Tc at z ¼ h=2; T ¼ Tm at z ¼ �h=2:

ð8Þ

The solution of Eq. (8) is obtained by means of a polynomial series[56] as:

TðzÞ ¼ Tm þ Tc � Tmð Þgðz;hÞ; ð9Þ

where

gðz;hÞ ¼ 1C

2zþ h2h

� �� jcm

ðnþ 1Þjm

2zþ h2h

� �nþ1"

þ j2cm

ð2nþ 1Þj2m

2zþ h2h

� �2nþ1

� j3cm

ð3nþ 1Þj3m

2zþ h2h

� �3nþ1

þ j4cm

ð4nþ 1Þj4m

2zþ h2h

� �4nþ1

� j5cm

ð5nþ 1Þj5m

2zþ h2h

� �5nþ1#;

ð10Þ

C ¼ 1� jcm

ðnþ 1Þjmþ j2

cm

ð2nþ 1Þj2m� j3

cm

ð3nþ 1Þj3mþ j4

cm

ð4nþ 1Þj4m

� j5cm

ð5nþ 1Þj5m; ð11Þ

where jcm = jc � jm and Tc, Tm denote the temperature of the cera-mic and metal phases, respectively.

Table 2Temperature dependent coefficient for material SI3N4/SUS304, Refs. [55,21].

Material Property Po P�1 P1 P2 P3

SI3N4 E (Pa) 348.43e9 0.0 3.070e�4 2.160e�7 �8.946e�11

a (1/K) 5.8723e�6 0.0 9.095e�4 0.0 0.0

SUS304 E (Pa) 201.04e9 0.0 3.079e�4 �6.534e�7 0.0a (1/K) 12.330e�6 0.0 8.086e�4 0.0 0.0

Table 3Non-dimensionalized natural frequency X ¼ xh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqm=Em

pfor a simply supported

square plate for ah ¼ 10, gradient index, n = 1. The material properties are assumed to

be the same as in [44,11]. The present formulation based on first order sheardeformation plate theory is compared with third order shear deformation theory(TSDT) [44] and higher order shear and normal deformable plate theory (HOSNDPT)[11].

Mode TSDT [44] HOSDNPT [11] FSDT (present)

1 0.0592 0.0584 0.06092 0.1428 0.1410 0.14223 0.1428 0.1410 0.1422

Table 4Non-dimensionalized natural frequency for a simply supported square plate with acenter crack for a

h ¼ 1000.

c/a Mode 1 Mode 2

Ref. [60] Ref. [61] Present Ref. [60] Ref. [61] Present

0.0 19.74 19.74 19.752 49.35 49.38 49.4100.2 19.38 19.40 19.399 49.16 49.85 49.3000.4 18.44 18.50 18.484 46.44 47.27 47.0180.6 17.33 17.37 17.439 37.75 38.92 38.928

2.2. Plate formulation

Using the Mindlin formulation, the displacements u, v, w at apoint (x,y,z) in the plate (see Fig. 1) from the medium surface areexpressed as functions of the mid-plane displacements uo, vo, wo

and indepedent rotations hx, hy of the normal in yz and xz planes,respectively, as:

uðx; y; z; tÞ ¼ uoðx; y; tÞ þ zhxðx; y; tÞ;vðx; y; z; tÞ ¼ voðx; y; tÞ þ zhyðx; y; tÞ;wðx; y; z; tÞ ¼ woðx; y; tÞ;

ð12Þ

where t is the time. The strains in terms of mid-plane deformationcan be written as:

e ¼ep

0

� þ

zeb

es

� : ð13Þ

The mid-plane strains ep, bending strain eb, shear strain es in Eq. (13)are written as:

ep ¼uo;x

vo;y

uo;y þ vo;x

8><>:

9>=>;; eb ¼

hx;x

hy;y

hx;y þ hy;x

8><>:

9>=>;; es ¼

hx þwo;x

hy þwo;y

� ;

ð14Þ

where the subscript ‘comma’ represents the partial derivative withrespect to the spatial coordinate succeeding it. The membranestress resultants N and the bending stress resultants M can be re-lated to the membrane strains, ep and bending strains eb throughthe following constitutive relations:

N ¼Nxx

Nyy

Nxy

8><>:

9>=>; ¼ Aep þ Beb � NT;

M ¼Mxx

Myy

Mxy

8><>:

9>=>; ¼ Bep þ Deb �MT;

ð15Þ

where the matrices a = Aij, B = Bij and D = Dij; (i, j = 1,2,6) are theextensional, bending-extensional coupling and bending stiffnesscoefficients and are defined as:

Table 5Comparison of non-dimensional linear frequencies of simply supported FGM plate (a/b = 1,a/h = 8) in thermal environment.

Temperature Tc, Tm Gradient n Mode 1 Mode 2

Ref. [62] Present Ref. [62] Present

Tm = 300, Tc = 400 0.0 12.397 12.311 29.083 29.0160.5 8.615 8.276 20.215 19.7721.0 7.474 7.302 17.607 17.3692.0 6.693 6.572 15.762 15.599

Tm = 300, Tc = 600 0.0 11.984 11.888 28.504 28.4210.5 8.269 7.943 19.784 19.3271.0 7.171 6.989 17.213 16.9592.0 6.398 6.269 15.384 15.207

Table 6Number of nodes and total number of degrees of freedom for FGM plates with andwithout crack used in the present analysis.

c/a Number of nodes Total degrees of freedom

0.0 225 5130.2 184 8400.4 264 12160.6 356 1604

Table 7Effect of aspect ratio, thickness, crack length and gradient index on the natural frequency of simply supported FGM plates in ambient temperature (Tc = 300 K,Tm = 300 K).

a/b a/h n Mode 1 Mode 2c/a c/a

0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6

1 10 0.0 18.357 17.815 16.805 15.829 43.923 43.316 39.765 31.6710.5 12.584 12.214 11.522 10.853 30.032 29.622 27.218 21.6971.0 11.069 10.742 10.132 9.544 26.371 26.008 23.888 19.0362.0 9.954 9.659 9.111 8.582 23.703 23.372 21.447 17.0775.0 9.026 8.759 8.262 7.782 21.527 21.222 19.455 15.479

10.0 8.588 8.334 7.861 7.405 20.511 20.221 18.538 14.74920 0.0 18.829 18.340 17.366 16.377 46.599 46.178 43.337 35.265

0.5 12.908 12.573 11.906 11.228 31.862 31.575 29.643 24.1331.0 11.353 11.058 10.470 9.873 27.983 27.730 26.028 21.1842.0 10.212 9.946 9.416 8.879 25.165 24.936 23.396 19.0335.0 9.263 9.021 8.541 8.054 22.868 22.658 21.252 17.281

10.0 8.813 8.584 8.127 7.664 21.786 21.586 20.247 16.465

2 10 0.0 43.821 39.836 33.381 28.147 67.409 67.127 65.149 48.0120.5 30.052 27.315 22.877 19.282 46.123 45.930 44.576 32.9011.0 26.433 24.023 20.116 16.953 40.510 40.341 39.153 28.8672.0 23.761 21.595 18.085 15.245 36.397 36.246 35.181 25.8885.0 21.532 19.572 16.399 13.830 33.026 32.889 31.924 23.450

10.0 20.485 18.622 15.606 13.163 31.460 31.329 30.409 22.34220 0.0 46.468 42.594 35.886 30.178 73.591 73.283 71.260 56.844

0.5 31.864 29.205 24.596 20.676 50.361 50.151 48.766 38.9201.0 28.031 25.684 21.621 18.172 44.251 44.067 42.846 34.1542.0 25.211 23.095 19.439 16.338 39.792 39.626 38.526 30.6675.0 22.860 20.943 17.634 14.826 36.131 35.980 34.982 27.826

10.0 21.747 19.927 16.784 14.114 34.408 34.264 33.315 26.513


Aij; Bij;Dij� �

¼Z h=2

�h=2Q ij 1; z; z2� �

dz: ð16Þ

The thermal stress resultant, NT and the moment resultant MT are:

NT ¼NT

xx

NTyy

NTxy

8>><>>:

9>>=>>; ¼

Z h=2

�h=2Q ijaðz; TÞ

110

8><>:

9>=>;DTðzÞdz;

MT ¼MT

xx

MTyy

MTxy

8>><>>:

9>>=>>; ¼

Z h=2

�h=2Qijaðz; TÞ

110

8><>:

9>=>;zDTðzÞdz;

ð17Þ

where the thermal coefficient of expansion a(z,T) is given by Eq. (5)and DT(z) = T(z) � To is the temperature rise from the referencetemperature To at which there are no thermal strains.

Similarly, the transverse shear force Q = {Qxz,Qyz} is related tothe transverse shear strains es through the following equation:

Q ij ¼ Eijes; ð18Þ

where Eij ¼R h=2�h=2 Qtitjdz; ði; j ¼ 4;5Þ is the transverse shear stiffness

coefficient, ti, tj is the transverse shear coefficient for non-uniformshear strain distribution through the plate thickness. The stiffnesscoefficients Qij are defined as:

Q 11 ¼ Q 22 ¼Eðz; TÞ1� m2 ; Q 12 ¼

mEðz; TÞ1� m2 ;

Q 16 ¼ Q 26 ¼ 0; Q 44 ¼ Q 55 ¼ Q 66 ¼Eðz; TÞ

2ð1þ mÞ ; ð19Þ

where the modulus of elasticity E(z,T) and Poisson’s ratio m are gi-ven by Eq. (4). The strain energy function U is given by

UðdÞ ¼ 12

ZX

e0pAep þ e0pBeb þ eTbBep þ e0bDeb þ e0sEes � eT

pN� eTbM

n odX;

ð20Þ

where d = {u,v,w,hx,hy} is the vector of the degree of freedom asso-ciated to the displacement field in a finite element discretization.

Following the procedure given in [57], the strain energy functionU given in Eq. (20) can be rewritten as:

UðdÞ ¼ 12

dTKd; ð21Þ

where K is the linear stiffness matrix. The kinetic energy of the plateis given by

TðdÞ ¼ 12

ZX

p _u2o þ _v2

o þ _w2o

�þ I _h2

x þ _h2y

� �n odX; ð22Þ

where p ¼R h=2�h=2 qðzÞdz; I ¼

R h=2�h=2 z2qðzÞdz and q(z) is the mass

density that varies through the thickness of the plate given by Eq.(6). The plate is subjected to a temperature field and this in turn

results in in-plane stress resultants ðNthxx;N

thyy;N

thxy Þ. The external

work due to the in-plane stress resultants Nthxx;N

thyy;N

thxy

� �developed

in the plate under the thermal load is:

VðdÞ ¼Z

X

12

Nthxx

owox

� �2

þ Nthyy

owoy

� �2

þ 2Nthxy

owox

� �owoy

� �" #(

þ h2

24Nth

xxohx

ox

� �2

þ ohy

ox

� �2( )

þ Nthyy

ohx

oy

� �2("

þ ohy

oy

� �2)

2Nthxy

ohx

oxohx

oyþ ohy

oxohy

oy

� �#)dX; ð23Þ

Substituting Eqs. (21)–(23) in Lagrange’s equations of motion, thefollowing governing equation is obtained:

M€dþ Kþ KGð Þd ¼ 0; ð24Þ

where M is the consistent mass matrix. After substituting the char-acteristic of the time function [49] €d ¼ �x2d, the following alge-braic equation is obtained:

Kþ KGð Þ �x2M �

d ¼ 0; ð25Þ

where K and KG are the stiffness matrix and geometric stiffness ma-trix due to thermal loads, respectively, x is the natural frequency.


The plate is subjected to a temperature field. So, the first step in thesolution process is to compute the in-plane stress resultants due tothe temperature field. These will then be used to compute the stiff-ness matrix of the system and then the frequencies are computedfor the system.

3. Element description and treatment of boundary conditions

3.1. Eight noded shear flexible element

The plate element employed here is a C0 continuous shear flex-ible element with five degrees of freedom (uo,vo,wo,hx,hy) at eightnodes in an 8-noded quadrilateral (QUAD-8) element. If the inter-polation functions for QUAD-8 are used directly to interpolate thefive variables (uo,vo,wo,hx,hy) in deriving the shear strains andmembrane strains, the element will lock and show oscillations inthe shear and membrane stresses. The field consistency requiresthat the transverse shear strains and membrane strains must beinterpolated in a consistent manner. Thus, the hx and hy terms inthe expressions for shear strain es have to be consistent with thederivative of the field functions, wo,x and wo,y. This is achieved byusing field redistributed substitute shape functions to interpolate

6

8

10

12

14

16

18

20

0 2 4 6 8 10

Mod

e 1

frequ

ency

, Ω 1

Gradient index, n

Crack Length, c/ac/a=0.0c/a=0.2c/a=0.4c/a=0.6

6

8

10

12

14

16

18

20

0 2 4 6 8 10

Mod

e 1

frequ

ency

, Ω 1

Gradient index, n

Crack Length, c/ac/a=0.0,c/a=0.2c/a=0.4c/a=0.6

Fig. 4. Frequency (mode 1) as a function of gradient index n and crack length c/a fora simply supported square FGM plate in ambient temperature (Tc = 300 K,Tm = 300 K).

those specific terms, which must be consistent as described in[48,49]. This element is free from locking syndrome and has goodconvergence properties. For complete description of the element,interested readers are referred to the literature [48,49], wherethe element behavior is discussed in great detail.

3.2. Skew boundary transformation

For skew plates supported on two adjacent edges, the edges ofthe boundary elements may not be parallel to the global axes(x,y,z). In order to specify the boundary conditions at skew edges,it is necessary to use edge displacements u0o;v 0o;w0o

�, etc., in local

coordinates (x0,y0,z0) (see Fig. 1).The element matrices correspond-ing to the skew edges are transformed from global axes to localaxes on which the boundary conditions can be conveniently spec-ified. The relation between the global and local degrees of freedomof a particular node can be obtained through simple transforma-tion rules [58] expressed as:

d ¼ Lgd0; ð26Þ

where di; d0i are the generalized displacement vectors in the global

and local coordinate system, respectively, of a node i and are de-fined as

10

15

20

25

30

35

40

45

0 2 4 6 8 10

Mod

e 2

frequ

ency

, Ω 2

Gradient index, n


15

20

25

30

35

40

45

50

0 2 4 6 8 10

Mod

e 2

frequ

ency

, Ω 2

Gradient index, n


Fig. 5. Frequency (mode 2) as a function of gradient index n and crack length c/afor a simply supported square FGM plate in ambient temperature (Tc = 300 K,Tm = 300 K).


d ¼ ½uo vo wo hx hy �T;

d0 ¼ u0o v 0o w0o h0x h0y� �T

;ð27Þ

The nodal transformation matrix for a node i, on the skew boundaryis given by

Lg ¼

cos w sin w 0 0 0� sin w cos w 0 0 0

0 0 1 0 00 0 0 cos w sin w

0 0 0 � sin w cos w

26666664

37777775; ð28Þ

where w is the angle of the plate. The transformation matrix for thenodes that do not lie on the skew edge is an identity matrix. Thus,for the complete element, the element transformation matrix iswritten as:

Te ¼ diag Lg Lg Lg Lg Lg Lg Lg Lgh i: ð29Þ

4. Numerical examples

In this section, we present the natural frequencies of crackedFGM plates. Fig. 2 shows the geometry and boundary condition

10

15

20

25

30

35

40

45

0 2 4 6 8 10

Mod

e 1

frequ

ency

, Ω 1

Gradient index, n


10

15

20

25

30

35

40

45

50

0 2 4 6 8 10

Mod

e 1

frequ

ency

, Ω 1

Gradient index, n


Fig. 6. Frequency (mode 1) as a function of gradient index n and crack length c/a fora simply supported rectangular FGM plate in ambient temperature (Tc = 300 K,Tm = 300 K).

of the plate. The crack is assumed to be at the center of the plate.For a rectangular plate, the crack is assumed to be parallel to thesides of length a. The effect of the gradient index n, temperaturegradient (Tc � Tm), crack length c, aspect ratio a/b, thickness ofthe plate h, skew angle of the plate w and boundary conditionson the natural frequency are numerically studied. The assumedvalues for the parameters are listed in Table 1.

In all cases, we present the non-dimensionalized free flexuralfrequency defined as:

X ¼ xa2

ffiffiffiffiffiffiffiffiqchD

r; ð30Þ

where x is the natural frequency, D ¼ Ec h3

12ð1�m2Þ is the bending rigidity

of a homogeneous ceramic plate, qc, Ec are the mass density andYoung’s modulus of the ceramic, respectively. Fig. 3 shows thevariation of the volume fractions of ceramic and metal, respectively,in the thickness direction z for the FGM plate. The top surface isceramic rich and the bottom surface is metal rich. The FGM plateconsidered here consists of silicon nitride (SI3N4) and stainless steel(SUS304). The material is considered to be temperature dependentand the temperature coefficients corresponding to SI3N4/SUS304 are

20

25

30

35

40

45

50

55

60

65

70

0 2 4 6 8 10

Mod

e 2

frequ

ency

, Ω 2

Gradient index, n


25

30

35

40

45

50

55

60

65

70

75

0 2 4 6 8 10

Mod

e 2

frequ

ency

, Ω 2

Gradient index, n


Fig. 7. Frequency (mode 2) as a function of gradient index n and crack length c/a fora simply supported rectangular FGM plate in ambient temperature (Tc = 300 K,Tm = 300 K).

Table 8Effect of aspect ratio, thickness, crack length and gradient index on the natural frequency of simply supported FGM plates with a temperature gradient (Tc = 400 K,Tm = 300 K).


0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6

1 10 0.0 17.962 17.433 16.465 15.537 43.382 42.803 39.331 31.3380.5 12.281 11.920 11.261 10.630 29.640 29.251 26.907 21.4581.0 10.786 10.468 9.889 9.337 26.014 25.672 23.606 18.8202.0 9.862 9.397 8.877 8.383 23.366 23.055 21.183 16.8765.0 8.753 8.495 8.028 7.584 21.196 20.912 19.199 15.285

10.0 8.309 8.065 7.623 7.204 20.177 19.909 18.282 14.55620 0.0 17.533 17.095 16.275 15.463 45.169 44.850 42.272 34.474

0.5 11.880 11.585 11.041 10.505 30.756 30.552 28.827 23.5291.0 10.379 10.122 9.651 9.189 26.947 26.772 25.267 20.6222.0 9.255 9.026 8.612 8.209 24.158 24.006 22.660 18.4915.0 8.276 8.073 7.713 7.365 21.843 21.714 20.507 16.735

10.0 7.785 7.596 7.266 6.949 20.730 20.614 19.483 15.905

2 10 0.0 43.280 39.345 32.999 27.855 66.733 66.461 64.522 47.6430.5 29.651 26.951 22.596 19.069 45.642 45.456 44.132 32.6471.0 26.067 23.690 19.860 16.760 40.077 39.915 38.754 28.6442.0 23.416 21.281 17.844 15.064 35.995 35.849 34.810 25.6855.0 21.196 19.266 16.166 13.657 32.638 32.507 35.168 23.262

10.0 20.149 18.317 15.374 12.992 31.073 30.949 30.057 22.16020 0.0 45.036 41.302 34.921 29.480 72.011 71.734 69.833 56.154

0.5 30.755 28.204 23.847 20.137 49.155 48.968 47.677 38.4121.0 26.992 24.746 20.921 17.667 43.129 42.965 41.833 33.6912.0 24.202 22.184 18.760 15.851 38.710 38.565 37.552 30.2305.0 21.835 20.018 16.948 14.337 35.044 34.914 34.006 27.400

10.0 20.691 18.974 16.079 13.614 33.295 33.175 32.320 26.087

Table 9Effect of aspect ratio, thickness, crack length and gradient index on the natural frequency of simply supported FGM plates with a temperature gradient (Tc = 600 K,Tm = 300 K).


0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6

1 10 0.0 17.105 16.605 15.730 14.910 42.272 41.754 38.451 30.6660.5 11.611 11.273 10.687 10.142 28.808 28.467 26.252 20.9581.0 10.155 9.859 9.350 8.878 25.241 24.944 23.000 18.3572.0 9.064 8.799 8.349 7.935 22.617 22.350 20.596 16.4295.0 8.115 7.788 7.483 7.123 20.425 20.187 18.598 14.828

10.0 7.642 7.420 7.054 6.723 19.371 19.150 17.654 14.07920 0.0 14.325 14.017 13.611 13.269 41.964 41.849 39.931 32.750

0.5 9.269 9.082 8.881 8.734 28.239 28.165 27.002 22.1901.0 7.866 7.712 7.573 7.492 24.564 24.501 23.545 19.3602.0 6.725 6.602 6.531 6.515 21.812 21.759 20.971 17.2545.0 5.543 5.458 5.483 5.565 19.401 19.358 18.755 15.455

10.0 4.818 4.761 4.864 5.023 18.161 18.125 17.645 14.564

2 10 0.0 42.172 38.336 32.221 27.270 65.381 65.129 63.276 46.9340.5 28.805 26.181 22.005 18.626 44.640 44.470 43.210 32.1321.0 25.277 22.971 19.309 16.348 39.151 39.003 37.902 28.1712.0 22.650 20.584 17.311 14.666 35.101 34.970 33.989 25.2345.0 20.412 18.553 15.621 13.252 31.722 31.607 30.730 22.805

10.0 19.335 17.575 14.808 12.571 30.118 30.010 29.182 21.68620 0.0 41.826 38.408 32.776 27.953 68.585 68.377 66.757 54.742

0.5 28.234 25.928 22.160 18.933 46.490 46.357 45.283 37.3381.0 24.608 22.593 19.324 16.530 50.619 40.506 39.579 32.6892.0 21.854 20.064 17.190 14.735 36.253 36.157 35.346 29.2605.0 19.387 17.809 15.318 13.185 32.503 32.427 31.732 26.409

10.0 18.115 16.651 14.370 12.410 30.636 30.573 29.943 25.056


listed in Table 2 [21,55]. The mass density (q) and thermalconductivity (K) are: qc = 2370 kg/m3, Kc = 9.19 W/mK for SI3N4

and qm = 8166 kg/m3, Km = 12.04 W/mK for SUS304. Poisson’s ratiom is assumed to be constant and taken as 0.28 for the current study[21,22]. Here, the modified shear correction factor obtained basedon energy equivalence principle as outlined in [59] is used. Theboundary conditions for simply supported and clamped cases are(see Fig. 2):

Simply supported boundary condition:

uo ¼ wo ¼ hy ¼ 0 on x ¼ 0; a;vo ¼ wo ¼ hx ¼ 0 on y ¼ 0; b:

ð31Þ

Clamped boundary condition:

uo ¼ wo ¼ hy ¼ vo ¼ hx ¼ 0 on x ¼ 0; a;uo ¼ wo ¼ hy ¼ vo ¼ hx ¼ 0 on y ¼ 0; b:

ð32Þ

00.20.40.60.810

0.5

1

0

0.1

0.2

0.3

0

0.5


Before proceeding with the detailed study on the effect of differentparameters on the natural frequency, the formulation developedherein is validated against available results pertaining to the linearfrequencies for functionally graded plate based on higher-ordershear deformation plate theory [44,11] and to the linear frequen-cies for cracked isotropic plates [60,61]. The computed frequenciesfor a square simply supported plate: (a) with gradient index n = 1and a/h = 10 is given in Table 3, (b) with a center crack and thick-ness a/h = 1000 are given in Table 4. It can be seen that the numer-ical results from the present formulation are found to be in goodagreement with the existing solutions.It is seen that with increase in crack length c/a, the natural fre-quency decreases. The decrease in frequency with increasing cracklength c/a is due to reduction in the stiffness of the plate. The linearfrequencies for uncracked FGM plates in thermal environment isshown in Table 5 and the results are compared with the resultsavailable in the literature [62]. Here the calculated non-dimen-sional frequency is defined as: X ¼ xða2=hÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqmð1� m2Þ=Em

p. Based

on the progressive mesh refinement, an 8 � 8 mesh is found to beadequate to model the full plate for the uncracked FGM plates. Themesh size used for cracked FGM plates and number of degrees offreedom are given in Table 6.

6

8

10

12

14

16

18

0 2 4 6 8 10

Mod

e 1

frequ

ency

, Ω 1

Gradient index, n

TemperatureTc=300,Tm=300,Tc=400,Tm=300Tc=600,Tm=300

15

20

25

30

35

40

45

0 2 4 6 8 10

Mod

e 2

frequ

ency

, Ω 2

Gradient index, n

TemperatureTc=300,Tm=300,Tc=400,Tm=300Tc=600,Tm=300

Fig. 8. Frequency as a function of gradient index n and different temperaturegradient for a simply supported rectangular FGM plate. The results are shown forcrack length c/a = 0.2.

Next, the linear flexural vibration behavior of FGM is numeri-cally studied with and without thermal environment. For theuniform temperature case, the material properties are evaluatedat Tc = Tm = T = 300 K. The variation of natural frequency with as-

0 0.2 0.4 0.6 0.8 1 0

0.5

1−0.5

Fig. 9. Mode shape for a square plate without a crack. The mode shape isnormalized with the maximum value.

00.20.40.60.81

0

0.5

1

0

0.1

0.2

0.3

00.20.40.60.81

00.20.40.60.81−0.5

0

0.5

Fig. 10. Mode shape for a simply supported square plate with a crack (c/a = 0.2,a/h = 10,Tc = 600,Tm = 300,n = 0). The mode shape is normalized with the maximumvalue.


pect ratio, thickness, gradient index and crack length is shown inTable 7 and graphically in Figs. 4–7. Figs. 4 and 5 shows the varia-tion of mode 1 and mode 2 frequency of a simply supported squareplate with gradient index for different crack length c/a and thick-ness h. Figs. 6 and 7 shows the variation of mode 1 and mode 2 fre-quency of a simply supported rectangular plate with gradientindex for different crack length c/a and thickness h. The natural fre-quency increases with decrease in the plate thickness and thisbehavior is maintained, regardless of the boundary conditions. Itis seen from Table 7 and Figs. 4–7 that with increase in crack lengthand gradient index, the linear frequency of the plate decreases andwith decrease in plate thickness, the natural frequency increases.The decrease in frequency with increasing crack length c/a is dueto the reduction in the stiffness of the plate and the decrease in fre-

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−0.2−0.1

00.10.2

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−0.2

0

0.2

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−0.2

0

0.2

Fig. 11. Mode shape (second mode) for a simply supported square plate withdifferent crack sizes (a/h = 10,Tc = 600,Tm = 300,n = 0). The mode shape is normal-ized with the maximum value.

quency with increasing gradient index n occurs due to increase inthe metallic volume fraction. In both the cases, the reduction in fre-quency is due to the stiffness degradation. It can also be observedthat the decrease in frequency is not proportional to the increase incrack length. The frequency decreases rapidly with increase incrack length. This observation is true, irrespective of thick or thin,and square or rectangular case considered in the present study.

The influence of thermal gradient on the linear frequencies ofFGMs is examined in Tables 8 and 9 and graphically in Fig. 8 withdifferent surface temperatures. The temperature field is assumedto vary only in the thickness direction and is determined by Eq.(9). The temperature for the ceramic surface is varied (Tc =400 K,600 K) while maintaining a constant value on the metallicsurface (Tm = 300 K). Figs. 9 and 10 shows the first two modeshapes of simply supported FGM plates with and without a crack.A similar trend can be observed for the FGM plates in thermal envi-ronment. The natural frequency decreases with increasing gradientindex and crack length. Also, the frequency further decreases withincrease in temperature gradient as expected. The mode shape(second mode) for a simply supported square plate with a centercrack is shown in Fig. 11 for different crack length and a/h = 10,Tc = 600, Tm = 300, n = 0. It is seen that with increase in cracklength, the plate becomes locally flexible.

Lastly, the effect of skewness of a simply supported andclamped square FGM plates in thermal environment is investi-gated. The thickness of the plate is taken as a/h = 10 and the plateis assumed to have a center crack c/a = 0.2. The results are shownin Tables 10 and 11 for various gradient indices and skew angles.It can be seen from Tables 10 and 11 that with increase in skewangle w, the linear frequency increases and with increase ingradient index n, the linear frequency decreases.

Table 10Influence of skew angle on linear frequencies for a simply supported FGM plate (a/b = 1,a/h = 10) with a center crack (c/a = 0.2) and with a temperature gradient,(Tc = 600 K,Tm = 300 K).

Skewangle,w

Frequency Gradient index, n

0.0 0.5 1.0 2.0 5.0 10.0

0� Mode 1 16.605 11.273 9.859 8.799 7.788 7.420Mode 2 41.754 28.467 24.944 22.350 20.187 19.150

15� Mode 1 18.351 12.480 10.918 9.751 8.748 8.257Mode 2 41.778 28.487 24.962 22.365 20.202 19.169

30� Mode 1 23.956 16.341 14.310 12.796 11.519 10.911Mode 2 47.407 32.365 28.365 25.416 22.969 21.814

45� Mode 1 35.793 24.473 21.448 19.200 17.325 16.450Mode 2 61.697 42.180 36.978 33.134 29.956 28.477

Table 11Influence of skew angle on linear frequencies for a clamped FGM plate (a/b = 1,a/h = 10) with a center crack (c/a = 0.2) and with a temperature gradient, (Tc = 600 K,Tm = 300 K).

Skewangle,w

Frequency Gradient index, n

0.0 0.5 1.0 2.0 5.0 10.0

0� Mode 1 29.331 19.991 17.501 15.666 14.147 13.431Mode 2 57.569 39.137 34.283 30.691 27.726 26.335

15� Mode 1 30.963 21.110 18.482 16.546 14.944 14.189Mode 2 57.338 39.178 34.318 30.724 27.755 26.382

30� Mode 1 36.649 25.010 21.903 19.611 17.717 16.831Mode 2 63.181 43.191 37.834 33.865 30.590 29.082

45� Mode 1 49.730 33.987 29.773 26.658 24.088 22.899Mode 2 78.840 53.951 106.69 42.281 38.179 36.311


5. Conclusion

The linear free flexural vibration behavior of FGM plates withand without thermal environment is numerically studied usingQUAD-8 shear flexible element. The formulation is based on firstorder shear deformation theory. The material is assumed to betemperature dependent and graded in the thickness directionaccording to the power-law distribution in terms of volumefractions of the constituents. The effective material properties areestimated using Mori–Tanaka homogenization method. Numericalexperiments have been conducted to bring out the effect ofgradient index, aspect ratio, crack length, thickness and boundarycondition on the natural frequency of the FGM plate. From thedetailed parametric study, it can be concluded that with increasein both the gradient index and crack length, the natural frequencydecreases. In both cases, the decrease is due to stiffnessdegradation.

Acknowledgements

S. Natarajan acknowledges the financial support of (1) theOverseas Research Students Awards Scheme; (2) the Faculty ofEngineering, University of Glasgow for period January 2008 –September 2009 and of (3) the School of Engineering (CardiffUniversity) for the period September 2009 onwards.

Stéphane Bordas and Pierre Kerfriden would like to acknowl-edge the financial support of the Royal Academy of Engineeringand of the Leverhulme Trust for Bordas’ Senior Research Fellowshipentitled ‘‘Towards the next generation surgical simulators’’ as well aspartial financial support of the UK Engineering and PhysicalScience Research Council(EPSRC) under grants EP/G069352/1Advanced discretisation strategies for ‘‘atomistic’’ nano CMOSsimulation and EP/G042705/1 Increased Reliability for IndustriallyRelevant Automatic Crack Growth Simulation with the eXtendedFinite Element Method.

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