Functionally Graded Materials

FunctionallyGraded

MaterialsNonlinear Analysis of

Plates and Shells

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FunctionallyGraded

MaterialsNonlinear Analysis of

Plates and Shells

Hui-Shen Shen

CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York

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Library of Congress Cataloging-in-Publication Data

Shen, Hui-Shen.Functionally graded materials : nonlinear analysis of plates and shells /

Hui-Shen Shen.p. cm.

Includes bibliographical references and index.ISBN 978-1-4200-9256-1 (alk. paper)1. Functionally gradient materials. 2. Shells (Engineering)--Thermal properties.

3. Plates (Engineering)--Thermal properties. I. Title.

TA418.9.F85S54 2009624.1776--dc22 2008036386

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Contents

Preface................................................................................................................... viiAuthor..................................................................................................................... ix

Chapter 1 Modeling of Functionally Graded Materialsand Structures ....................................................................... 1

1.1 Introduction ................................................................................................... 11.2 Effective Material Properties of FGMs....................................................... 31.3 Reddys Higher Order Shear Deformation Plate Theory........................ 91.4 Generalized Krmn-Type Nonlinear Equations ................................... 14References.............................................................................................................. 17

Chapter 2 Nonlinear Bending of Shear Deformable FGM Plates ... 212.1 Introduction ................................................................................................. 212.2 Nonlinear Bending of FGM Plates under Mechanical

Loads in Thermal Environments .............................................................. 222.3 Nonlinear Thermal Bending of FGM Plates

due to Heat Conduction............................................................................. 36References.............................................................................................................. 42

Chapter 3 Postbuckling of Shear Deformable FGM Plates ............. 453.1 Introduction ................................................................................................. 453.2 Postbuckling of FGM Plates with Piezoelectric Actuators

under Thermoelectromechanical Loads................................................... 473.3 Thermal Postbuckling Behavior of FGM Plates

with Piezoelectric Actuators ...................................................................... 663.4 Postbuckling of Sandwich Plates with FGM Face Sheets

in Thermal Environments .......................................................................... 82References.............................................................................................................. 96

Chapter 4 Nonlinear Vibration of Shear DeformableFGM Plates .......................................................................... 99

4.1 Introduction ................................................................................................. 994.2 Nonlinear Vibration of FGM Plates in Thermal Environments ......... 100

4.2.1 Free Vibration ................................................................................ 1084.2.2 Forced Vibration............................................................................ 108

4.3 Nonlinear Vibration of FGM Plates with Piezoelectric Actuatorsin Thermal Environments ........................................................................ 118

4.4 Vibration of Postbuckled Sandwich Plates with FGMFace Sheets in Thermal Environments ................................................... 132

References............................................................................................................ 143

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Chapter 5 Postbuckling of Shear Deformable FGM Shells ........... 1475.1 Introduction ............................................................................................... 1475.2 Boundary Layer Theory for the Buckling of FGM

Cylindrical Shells....................................................................................... 1495.2.1 Donnell Theory.............................................................................. 1495.2.2 Generalized KrmnDonnell-Type Nonlinear Equations ..... 1505.2.3 Boundary Layer-Type Equations ................................................ 152

5.3 Postbuckling Behavior of FGM Cylindrical Shellsunder Axial Compression ........................................................................ 153

5.4 Postbuckling Behavior of FGM Cylindrical Shellsunder External Pressure ........................................................................... 169

5.5 Postbuckling Behavior of FGM Cylindrical Shellsunder Torsion ............................................................................................ 183

5.6 Thermal Postbuckling Behavior of FGMCylindrical Shells....................................................................................... 199

References............................................................................................................ 208

Appendix A............................................................................................ 211

Appendix B ............................................................................................ 213

Appendix C ............................................................................................ 215

Appendix D ........................................................................................... 221

Appendix E ............................................................................................ 225

Appendix F ............................................................................................ 227

Appendix G ........................................................................................... 229

Appendix H ........................................................................................... 231

Appendix I ............................................................................................. 233

Appendix J ............................................................................................. 235

Appendix K............................................................................................ 239

Appendix L ............................................................................................ 241

Appendix M ........................................................................................... 245

Appendix N ........................................................................................... 247

Appendix O ........................................................................................... 257

Index ....................................................................................................... 259

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vi Contents

Preface

With the development of new industries and modern processes, many struc-tures serve in thermal environments, resulting in a new class of compositematerials called functionally graded materials (FGMs). FGMs were initiallydesigned as thermal barrier materials for aerospace structural applicationsand fusion reactors. They are now developed for general use as structuralcomponents in extremely high-temperature environments. The ability topredict the response of FGM plates and shells when subjected to thermaland mechanical loads is of prime interest to structural analysis. In fact, manystructures are subjected to high levels of load that may result in nonlinearloaddeection relationships due to large deformations. One of the impor-tant problems deserving special attention is the study of their nonlinearresponse to large deection, postbuckling, and nonlinear vibration.This book consists of ve chapters. The chapter and section titles are

signicant indicators of the content matter. Each chapter contains adequateintroductory material to enable engineering graduates who are familiar withthe basic understanding of plates and shells to follow the text. The modelingof FGMs and structures is introduced and the derivation of the governingequations of FGM plates in the von Krmn sense is presented in Chapter 1.In Chapter 2, the geometrically nonlinear bending of FGM plates due totransverse static loads or heat conduction is presented. Chapter 3 furnishesa detailed treatment of the postbuckling problems of FGM plates subjected tothermal, electrical, and mechanical loads. Chapter 4 deals with the nonlinearvibration of FGM plates with or without piezoelectric actuators. Finally,Chapter 5 presents postbuckling solutions for FGM cylindrical shells undervarious loading conditions. Most of the solutions presented in these chaptersare the results of investigations conducted by the author and his collabor-ators since 2001. The results presented herein may be treated as a benchmarkfor checking the validity and accuracy of other numerical solutions.Despite a number of existing texts on the theory and analysis of plates

and=or shells, there is not a single book that is devoted entirely to thegeometrically nonlinear problems of inhomogeneous isotropic and function-ally graded plates and shells. It is hoped that this book will ll the gap tosome extent and be used as a valuable reference source for postgraduatestudents, engineers, scientists, and applied mathematicians in this eld.I wish to recordmy appreciation to the National Natural Science Foundation

of China (grant nos. 59975058 and 50375091) for partially funding this work,and I also wish to thank my wife for her encouragement and forbearance.

Hui-Shen Shen

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vii

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Author

Hui-Shen Shen is a professor of applied mechanics at Shanghai Jiao TongUniversity. He graduated from Tsinghua University in 1970, and received hisMSc in solid mechanics and his PhD in structural mechanics from ShanghaiJiao Tong University in 1982 and 1986, respectively. In 19911992, he wasa visiting research fellow at the University of Wales (Cardiff) and theUniversity of Liverpool in the United Kingdom. Dr. Shen became a fullprofessor of applied mechanics at Shanghai Jiao Tong University at the endof 1992. In 1995, he was invited again as a visiting professor at the Universityof Cardiff and in 19981999, as a visiting research fellow at the Hong KongPolytechnic University, and in 20022003 as a visiting professor at the CityUniversity of Hong Kong. Also in 2002, he was a Tan Chin Tuan exchangefellow at the Nanyang Technological University in Singapore and in 2004, hewas a Japan Society for the Promotion of Science (JSPS) invitation fellow atthe Shizuoka University in Japan. In 2007, Dr. Shen was a visiting professorat the University of Western Sydney in Australia.Dr. Shens research interests include stability theory and, in general, non-

linear response of plate and shell structures. He has published over 190 journalpapers, of which 123 are international journal papers. His research publica-tions have been widely cited in the areas of computational mechanics andstructural engineering (more than 1500 times by papers published in 387 inter-national archival journals, and 220 local journals, excluding self-citations).He is the coauthor of the books Buckling of Structures (with T.-Y. Chen) andPostbuckling Behavior of Plates and Shells. He won the second Science andTechnology Progress Awards of Shanghai in 1998 and 2003, respectively.Currently, Dr. Shen serves on the editorial boards of the journal AppliedMathematics and Mechanics (ISSN: 0253-4827) and the International Journal ofStructural Stability and Dynamics (ISSN: 0219-4554). He is a member of theAmerican Society of Civil Engineers.

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1Modeling of Functionally Graded Materialsand Structures

1.1 Introduction

The most lightweight composite materials with high strength=weight andstiffness=weight ratios have been used successfully in aircraft industry andother engineering applications. However, the traditional composite materialis incapable to employ under the high-temperature environments. In general,the metals have been used in engineering eld for many years on account oftheir excellent strength and toughness. In the high-temperature condition,the strength of the metal is reduced similar to the traditional compositematerial. The ceramic materials have excellent characteristics in heat resist-ance. However, the applications of ceramic are usually limited due to theirlow toughness.Recently, a new class of composite materials known as functionally graded

materials (FGMs) has drawn considerable attention. A typical FGM, with ahigh bendingstretching coupling effect, is an inhomogeneous compositemade from different phases of material constituents (usually ceramic andmetal). An example of such material is shown in Figure 1.1 (Yin et al. 2004)where spherical or nearly spherical particles are embedded within an iso-tropic matrix. Within FGMs the different microstructural phases have differ-ent functions, and the overall FGMs attain the multistructural status fromtheir property gradation. By gradually varying the volume fraction of con-stituent materials, their material properties exhibit a smooth and continuouschange from one surface to another, thus eliminating interface problems andmitigating thermal stress concentrations. This is due to the fact that theceramic constituents of FGMs are able to withstand high-temperature envir-onments due to their better thermal resistance characteristics, while the metalconstituents provide stronger mechanical performance and reduce the pos-sibility of catastrophic fracture.The term FGMs was originated in the mid-1980s by a group of scientists in

Japan (Yamanoushi et al. 1990, Koizumi 1993). Since then, an effort todevelop high-resistant materials using FGMs had been continued. FGMswere initially designed as thermal barrier materials for aerospace structuresand fusion reactors (Hirai and Chen 1999, Chan 2001, Uemura 2003). They

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1

are now developed for the general use as structural components in high-temperature environments. An example is FGM thin-walled rotating bladesas shown in Figure 1.2 (Librescu and Song 2005). Potential applications ofFGM are both diverse and numerous. Applications of FGMs have recentlybeen reported in the open literature, e.g., FGM sensors (Mller et al. 2003)and actuators (Qiu et al. 2003), FGM metal=ceramic armor (Liu et al. 2003),FGM photodetectors (Paszkiewicz et al. 2008), and FGM dental implant(Watari et al. 2004, see Figure 1.3). A number of reviews dealing with various

FIGURE 1.1An FGM with the volume fractions ofconstituent phases graded in one (verti-cal) direction. (FromYin, H.M., Sun, L.Z.,and Paulino, G.H., Acta Mater., 52, 3535,2004. With permission.)

Phase B particleswith

phase A matrix

Phase A particleswith

phase B matrix

Transition zone

Y

yD

y

xz, Z

X

xpr1

Ro

L

g

g +bod

FIGURE 1.2An FGM thin-walled tapered pretwisted turbine blade. (From Librescu, L. and Song, S.-Y.,J. Therm. Stresses, 28, 649, 2005. With permission.)


2 Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

aspects of FGMs have been published in the past few decades (Fuchiyamaand Noda 1995, Markworth et al. 1995, Tanigawa 1995, Noda 1999, Paulinoet al. 2003). They show that most of early research studies in FGMs had morefocused on thermal stress analysis and fracture mechanics. A comprehensivesurvey for bending, buckling, and vibration analysis of plate and shellstructures made of FGMs was presented by Shen (2004). Recently, Birmanand Byrd (2007) presented a review of the principal developments in FGMsthat includes heat transfer issues, stress, stability and dynamic analyses,testing, manufacturing and design, applications, and fracture.

1.2 Effective Material Properties of FGMs

Several FGMs are manufactured by two phases of materials with differentproperties. A detailed description of actual graded microstructures is usuallynot available, except perhaps for information on volume fraction distribu-tion. Since the volume fraction of each phase gradually varies in the grad-ation direction, the effective properties of FGMs change along this direction.Therefore, we have two possible approaches to model FGMs. For the rstchoice, a piecewise variation of the volume fraction of ceramic or metal isassumed, and the FGM is taken to be layered with the same volume fractionin each region, i.e., quasihomogeneous ceramicmetal layers (Figure 1.4a).For the second choice, a continuous variation of the volume fraction ofceramic or metal is assumed (Figure 1.4b), and the metal volume fractioncan be represented as the following function of the thickness coordinate Z.

Vm 2Z h2h N

(1:1)

where h is the thickness of the structure, and N (0 N 1) is a volumefraction exponent, which dictates the material variation prole through the

2 mm

FIGURE 1.3Ti=20HAP FGM dental implant. External appearance (left) and cross-section(right). (FromWatari, F., Yokoyama,A., Omori,M., Hirai, T., Kondo,H., Uo,M.,and Kawasaki, T., Compos. Sci. Technol., 64, 893, 2004. With permission.)


Modeling of Functionally Graded Materials and Structures 3

FGM layer thickness. As is presented in Figure 1.5, changing the value ofN generates an innite number of composition distributions.In order to accurately model the material properties of FGMs, the proper-

ties must be temperature- and position-dependent. This is achieved by using

(a) (b)

FIGURE 1.4Analytical model for an FGM layer.

0.500.00

0.25

0.50

0.75

1.00

N= 10.0N= 5.0

N= 3.0N= 2.0

N= 1.0

N= 0.5

N= 0.2

N= 0.1

V m

Z/h

0.25 0.00 0.25 0.50

N= 100

FIGURE 1.5Volume fraction of metal along the thickness.



a simple rule of mixture of composite materials (Voigt model). The effectivematerial properties Pf of the FGM layer, like Youngs modulus Ef, andthermal expansion coefcient af, can then be expressed as

Pf Xj1

PjVfj (1:2)

where Pj and Vfj are the material properties and volume fraction of theconstituent material j, and the sum of the volume fractions of all the con-stituent materials makes 1, i.e., X

j1Vfj 1 (1:3)

Since functionally graded structures are most commonly used in high-temperature environment where signicant changes in mechanical propertiesof the constituent materials are to be expected (Reddy and Chin 1998), it isessential to take into consideration this temperature-dependency for accurateprediction of the mechanical response. Thus, the effective Youngs modulus Ef,Poissons ratio nf, thermal expansion coefcient af, and thermal conductivitykf are assumed to be temperature dependent and can be expressed as a non-linear function of temperature (Touloukian 1967):

Pj P0(P1T1 1 P1T P2T2 P3T3) (1:4)

where P0, P1, P1, P2, and P3 are the coefcients of temperature T (in K) andare unique to the constituent materials. Typical values for Youngs modulusEf (in Pa), Poissons ratio nf, thermal expansion coefcient af (in K

1), and thethermal conductivity kf (in W mK

1) of ceramics and metals are listed inTables 1.1 through 1.4 (from Reddy and Chin 1998). From Equations 1.1through 1.3, one has (Gibson et al. 1995):

Ef(Z,T) [Em(T) Ec(T)] 2Z h2h N

Ec(T) (1:5a)

TABLE 1.1

Modulus of Elasticity of Ceramics and Metals in Pa for Ef

Materials P0 P1 P1 P2 P3

Zirconia 244.27e 9 0 1.371e 3 1.214e 6 3.681e 10Aluminum oxide 349.55e 9 0 3.853e 4 4.027e 7 1.673e 10Silicon nitride 348.43e 9 0 3.070e 4 2.160e 7 8.946e 11Ti-6Al-4V 122.56e 9 0 4.586e 4 0 0Stainless steel 201.04e 9 0 3.079e 4 6.534e 7 0Nickel 223.95e 9 0 2.794e 4 3.998e 9 0Source: Reddy, J.N. and Chin, C.D., J. Therm. Stresses, 21, 593, 1998. With permission.



TABLE 1.2

Coefcient of Thermal Expansion of Ceramics and Metals in K1 for af


Zirconia 12.766e 6 0 1.491e 3 1.006e 5 6.778e 11Aluminum oxide 6.8269e 6 0 1.838e 4 0 0Silicon nitride 5.8723e 6 0 9.095e 4 0 0Ti-6Al-4V 7.5788e 6 0 6.638e 4 3.147e 6 0Stainless steel 12.330e 6 0 8.086e 4 0 0Nickel 9.9209e 6 0 8.705e 4 0 0Source: Reddy, J.N. and Chin, C.D., J. Therm. Stresses, 21, 593, 1998. With permission.

TABLE 1.3

Thermal Conductivity of Ceramics and Metals in W mK1 for kf


Zirconia 1.7000 0 1.276e 4 6.648e 8 0Aluminum oxide 14.087 1123.6 6.227e 3 0 0Silicon nitride 13.723 0 1.032e 3 5.466e 7 7.876e 11Ti-6Al-4V 1.0000 0 1.704e 2 0 0Stainless steel 15.379 0 1.264e 3 2.092e 6 7.223e 10Nickela 187.66 0 2.869e 3 4.005e 6 1.983e 9Nickelb 58.754 0 4.614e 4 6.670e 7 1.523e 10Source: Reddy, J.N. and Chin, C.D., J. Therm. Stresses, 21, 593, 1998. With permission.a For 300 K T 635 K.b For 635 K T.

TABLE 1.4

Poissons Ratio of Ceramics and Metals for nf


Zirconia 0.2882 0 1.133e 4 0 0Aluminum oxide 0.2600 0 0 0 0

Silicon nitride 0.2400 0 0 0 0

Ti-6Al-4V 0.2884 0 1.121e 4 0 0Stainless steel 0.3262 0 2.002e 4 3.797e 7 0Nickel 0.3100 0 0 0 0

Source: Reddy, J.N. and Chin, C.D., J. Therm. Stresses, 21, 593, 1998. With permission.



af(Z,T) [am(T) ac(T)] 2Z h2h N

ac(T) (1:5b)

kf(Z,T) [km(T) kc(T)] 2Z h2h N

kc(T) (1:5c)

nf(Z,T) [nm(T) nc(T)] 2Z h2h N

nc(T) (1:5d)

It is evident that Ef, nf, af, and kf are both temperature- and position-dependent.This method is simple and convenient to apply for predicting the overallmaterial properties and responses; however, owing to the assumed simplica-tions the validity is affected by the detailed graded microstructure.As argued before, precise information about the size, the shape, and the

distribution of particles is not available and the effective elastic moduli of thegraded microstructures must be evaluated based on the volume fractiondistribution and the approximate shape of the dispersed phase. Severalmicromechanics models have also been developed over the years to inferthe effective properties of FGMs. The MoriTanaka scheme (Mori andTanaka 1973, Benveniste 1987) for estimating the effective moduli is applic-able to regions of the graded microstructure which have a well-denedcontinuous matrix and a discontinuous particulate phase as depicted inFigure 1.1. It takes into account the interaction of the elastic elds amongneighboring inclusions. It is assumed that the matrix phase, denoted by thesubscript 1, is reinforced by spherical particles of a particulate phase, denotedby the subscript 2. In this notation, K1, G1, and V1 denote, respectively, thebulk modulus, the shear modulus, and the volume fraction of the matrixphase; K2, G2, and V2 denote the corresponding material properties and thevolume fraction of the particulate phase. It should be noted that V1V2 1.The effective local bulk modulus Kf, the shear modulus Gf, thermal expansioncoefcient af, and thermal conductivity kf obtained by the MoriTanakascheme for a random distribution of isotropic particles in an isotropic matrixare given by

Kf K1K2 K1

V21 (1 V2) 3(K2 K1)=(3K1 4G1) (1:6a)

Gf G1G2 G1

V21 (1 V2) (G2 G1)=(G1 f1) (1:6b)

af a1a2 a1

(1=Kf) (1=K1)(1=K2) (1=K1) (1:6c)

kf k1k2 k1

V21 (1 V2) (k2 k1)=3k1 (1:6d)



where

f1 G1(9K1 8G1)6(K1 2G1) (1:7)

The self-consistent method (Hill 1965) assumes that each reinforcementinclusion is embedded in a continuum material whose effective propertiesare those of the composite. This method does not distinguish between matrixand reinforcement phases and the same overall moduli are predicted inanother composite in which the roles of the phases are interchanged. Thismakes it particularly suitable for determining the effective moduli in thoseregions which have an interconnected skeletal microstructure as depicted inFigure 1.6. The locally effective elastic moduli by the self-consistent methodare given by

d

Kf V1Kf K2

V2Kf K1 (1:8a)

h

Gf V1Gf G2

V2Gf G1 (1:8b)

where

d 3 5h KfKf (4=3)Gf (1:9)

FIGURE 1.6Skeletal microstructure of FGM material. (From Vel, S.S. and Batra, R.C., AIAA J., 40, 1421, 2002.With permission.)



From Equation 1.8a, one has

Kf 1V1=(K1 (4=3)Gf) V2=(K2 (4=3)Gf) 43Gf (1:10)

and Gf is obtained by solving the following quartic equation:

V1K1=(K1 4Gf=3) V2K2=(K2 4Gf=3) 5 V1G2=(Gf G2) V2G1=(Gf G1) 2 0 (1:11)

Then, the effective Youngs modulus Ef and Poissons ratio nf can be foundfrom Ef 9KfGf=3KfGf and nf (3Kf 2Gf)=2(3KfGf), respectively.A comparison between the MoriTanaka and self-consistent models and

the nite element simulation of FGM was presented in Reuter et al. (1997)and Reuter and Dvorak (1998). The MoriTanaka model was shown to yieldaccurate prediction of the properties with a well-dened continuous matrixand discontinuous inclusions, while the self-consistent model was better inskeletal microstructures characterized by a wide transition zone between theregions with predominance of one of the constituent phases.

1.3 Reddys Higher Order Shear Deformation Plate Theory

Reddy (1984a,b) developed a simple higher order shear deformation platetheory (HSDPT), in which the transverse shear strains are assumed to beparabolically distributed across the plate thickness. The theory is simple inthe sense that it contains the same dependent unknowns as in the rst-ordershear deformation plate theory (FSDPT), and no shear correction factors arerequired.Consider a rectangular plate made of FGMs. The length, width, and total

thickness of the plate are a, b, and h. As usual, the coordinate system has itsorigin at the corner of the plate on the midplane. Let U, V, andW be the platedisplacements parallel to a right-hand set of axes (X,Y,Z), where X islongitudinal and Z is perpendicular to the plate. Cx and Cy are the midplanerotations of the normal about the Y and X axes, respectively. The displace-ment components are assumed to be of the following form:

U1 U(X,Y, t) ZCx(X,Y, t) Z2jx(X,Y, t) Z3zx(X,Y, t) (1:12a)U2 V(X,Y, t) ZCy(X,Y, t) Z2jy(X,Y, t) Z3zy(X,Y, t) (1:12b)U3 W(X,Y, t) (1:12c)



where t represents time, U, V, W, Cx, Cy, jx, jy, zx, and zy are unknowns.If the transverse shear stresses s4 and s5 are to vanish at the bounding

planes of the plate (at Zh=2), the transverse shear strains 4 and 5 shouldalso vanish there. That is

5 X,Y, h2 , t

0, 4 X,Y, h2 , t

0 (1:13)

which imply the following conditions

jx 0 (1:14a)jy 0 (1:14b)

zx 43h2

@W@X

Cx

(1:14c)

zy 43h2

@W@Y

Cy

(1:14d)

Putting the above conditions in Equation 1.12 leads to the following displace-ment eld

U1 U 2 Cx x 432h

2Cx @W

@X

" #(1:15a)

U2 V 2 Cy x 432h

2Cy @W

@Y

" #(1:15b)

U3 W (1:15c)

in which x is a tracer. If x 1, Equation 1.15 is for the case of the HSDPT,which contains the same dependent unknowns (U, V, W, Cx, and Cy) as inthe FSDPT. If x 0, Equation 1.15 is reduced to the case of the FSDPT.The strains of the plate associated with the displacement eld given in

Equation 1.15 are

1 01 Z k01 Z2k21

2 02 Z k02 Z2k22

3 04 04 Z2k245 05 Z2k256 06 Z k06 Z2k26

(1:16)



where

01 @U@X

12

@W@X

2, k01

@Cx@X

, k21 x43h2

@Cx@X

@2W

@X2

02 @V@Y

12

@W@Y

2, k02

@Cy

@Y, k22 x

43h2

@Cy

@Y @

2W@Y2

!

04 Cy @W@Y

, k24 x4h2

Cy @W@Y

05 Cx @W@X

, k25 x4h2

Cx @W@X

06 @U@Y

@V@X

@W@X

@W@Y

k06 @Cx@Y

@Cy@X

k26 x43h2

@Cx@Y

@Cy@X

2 @2W

@X@Y

!

(1:17)

The plane stress constitutive equations may then be written in the form:

s1s2s6

24

35 Q11 Q12 0Q21 Q22 0

0 0 Q66

24

35 12

6

24

35 (1:18a)

s4s5

Q44 00 Q55

45

(1:18b)

where Qij are the transformed reduced stiffnesses dened by

Q11 Q22 Ef(Z,T)1 n2f, Q12 nfEf(Z,T)1 n2f

,

Q16 Q26 0, Q44 Q55 Q66 Ef(Z,T)2(1 nf)(1:19)

As in the classical plate theory, the stress resultants and couples are dened by

(Ni,Mi,Pi) h=2

h=2

si(1,Z,Z3)dZ, i 1, 2, 6 (1:20a)

(Q2,R2) h=2

h=2

s4(1,Z2)dZ (1:20b)



(Q1,R1) hh

s5(1,Z2)dZ (1:20c)

whereNi and Qi are the membrane and transverse shear forcesMi is the bending moment per unit lengthPi and Ri are the higher order bending moment and shear force,respectively

Substituting Equation 1.18 into Equation 1.20, and taking Equation 1.16into account, yields the constitutive relations of the plate

NMP

24

35 A B EB D F

E F H

24

35 0k0

k2

24

35 (1:21a)

QR

A D

D F

0

k2

(1:21b)

where Aij, Bij, etc. are the plate stiffnesses, dened by

(Aij,Bij,Dij,Eij,Fij,Hij) h=2

h=2

(Qij)(1,Z,Z2,Z3,Z4,Z6)dZ, i, j 1, 2, 6 (1:22a)

(Aij,Dij,Fij) h=2

h=2

(Qij)(1,Z2,Z4)dZ, i, j 4, 5 (1:22b)

The Hamilton principle for an elastic body is

t2t1

(dU dV dK)dt 0 (1:23)

wheredU is the virtual strain energydV is the virtual work done by external forcesdK is the virtual kinetic energy

dU V

h=2h=2

(sidi)dZdX dY

V

(Nid0i Midk0i Pidk2i )dZdX dY, i 1, 2, 6 (1:24a)



dV V

[q(X,Y)dU3]dX dY (1:24b)

dK V

h=2h=2

r( _Ujd _Uj)dZdX dY, j 1, 2, 3 (1:24c)

In Equation 1.24c, the superposed dots indicate differentiation with respect totime. Integrating Equation 1.23, and collecting the coefcients of dU, dV, dW,dCx, and dCy, we obtain the following equations of motion

dU:@N1@X

@N6@Y

I1 @2U@t2

I2 @2Cx@t2

c1I4 @3W

@X@t2

dV:@N6@X

@N2@Y

I1 @2V@t2

I2@2Cy

@t2 c1I4 @

3W@Y@t2

dW:@Q1@X

@Q2@Y

@@X

N1@W@X

N6 @W@Y

@

@YN6

@W@X

N2 @W@Y

q c2 @R1@X

@R2@Y

c1 @

2P1@X2

2 @2P6

@X@Y @

2P2@Y2

I1 @2W@t2

c21I7@2

@t2@2W@X2

@2W@Y2

c1I4 @

2

@t2@U@X

@V@Y

c1I5 @

2

@t2@Cx@X

@Cy@Y

!

dCx:@M1@X

@M6@Y

Q1 C2R1 C1@P1@X

@P6@Y

I2 @

2U@t2

I3 @2Cx@t2

c1I5 @3W

@X@t2

dCy:@M6@X

@M2@Y

Q2 c2R2 c1@P6@X

@P2@Y

I2 @

2V@t2

I3@2Cy

@t2 c1I5 @

3W@Y@t2

(1:25)

where c1 4=3h2, c2 3c1, and

I2 I2 c1I4, I5 I5 c1I7, I3 I3 2c1I5 c21I7, I8 I3 I5 (1:26a)

and the inertias Ii (i 1, 2, 3, 4, 5, 7) are dened by

(I1, I2, I3, I4, I5, I7) h=2

h=2

r(Z)(1,Z,Z2,Z3,Z4,Z6)dZ (1:26b)

where r is the mass density of the plate, which may also be positiondependent.



1.4 Generalized Krmn-Type Nonlinear Equations

Based on Reddys HSDPT with a von Krmn-type of kinematic nonlinearity(Reddy 1984b) and including thermal effects, Shen (1997) derived a set ofgeneral von Krmn-type equations which can be expressed in terms of astress function F, two rotations Cx and Cy, and a transverse displacementW,along with the initial geometric imperfection W*. These equations are thenextended to the case of shear deformable FGM plates.Let F (X,Y) be the stress function for the stress resultants dened by

Nx F,yy, Ny F,xx, and NxyF,xy, where a comma denotes partial differ-entiation with respect to the corresponding coordinates.If thermal effect is taken into account, we assume

N* N NT, M* M MT, P* P PT (1:27)

where NT, MT, ST, and PT are the forces, moments, and higher ordermoments caused by elevated temperature, and are dened by

NTx M

Tx P

Tx

NTy M

Ty P

Ty

NTxy M

Txy P

Txy

26664

37775

h=2h=2

AxAyAxy

264

375(1,Z,Z3)DT(X,Y,Z)dZ (1:28a)

STx

STy

STxy

26664

37775

MTx

MTy

MTxy

26664

37775 43h2

PTx

PTy

PTxy

26664

37775 (1:28b)

where DT(X,Y,Z)T(X,Y,Z)T0 is temperature rise from the referencetemperature T0 at which there are no thermal strains, and

AxAyAxy

24

35 Q11 Q12 Q16Q12 Q22 Q26

Q16 Q26 Q66

24

35 1 00 1

0 0

24

35 a11

a22

(1:29)

where a11 and a22 are the thermal expansion coefcients measured in thelongitudinal and transverse directions, respectively.The partial inverse of Equation 1.21a yields

0

M*P*

24

35 A* B* E*(B*)T D* (F*)T

(E*)T F* H*

24

35 N*k0

k2

24

35 (1:30)



where the superscript T represents the matrix transpose and in which thereduced stiffness matrices [Aij*], [Bij*], [Dij*], [Eij*], [Fij*], and [Hij*] (i,j 1, 2, 6) arefunctions of temperature and position, determined through relationships(Shen 1997):

A* A1, B* A1B, D* D BA1B, E* A1E,F* F EA1B, H* H EA1E (1:31)

From Equation 1.30, the bending moments, higher order moments, andtransverse shear forces can be written in the form:

Mx M1 B11* F,yy B21* F,xx D11* Cx,x D12* Cy,y c1[F11* (Cx,x W,xx) F21* (Cy,y W,yy)]MTx (1:32a)

My M2 B12* F,yy B22* F,xx D12* Cx,x D22* Cy,y c1[F12* (Cx,x W,xx) F22* (Cy,y W,yy)]MTy (1:32b)

Mxy M6 B66* F,xy D66* (Cx,y Cy,x) c1F66* (Cx,y Cy,x 2W,xy)MTxy (1:32c)

Px P1 E11* F,yy E21* F,xx F11* Cx,x F12* Cy,y c1[H11* (Cx,x W,xx)H12* (Cy,y W,yy)] PTx (1:32d)

Py P2 E12* F,yy E22* F,xx F21* Cx,x D22* Cy,y c1[H12* (Cx,x W,xx)H22* (Cy,y W,yy)] PTy (1:32e)

Q1 (A55 c2D55)(Cx W,x) (1:32f)R1 (D55 c2F55)(Cx W,x) (1:32g)Q2 (A44 c2D44)(Cy W,y) (1:32h)R2 (D44 c2F44)(Cy W,y) (1:32i)

Substituting Equation 1.32 into Equation 1.25, and considering the condi-tion of compatibility, which is also expressed in terms of F, Cx, Cy, W, andW*, the equations of motion are obtained in the following

~L11(W) ~L12(Cx) ~L13(Cy) ~L14(F) ~L15(NT) ~L16(MT) ~L(W W*, F) ~L17( W) I8( Cx,x Cy,y) q (1:33)

~L21(F) ~L22(Cx) ~L23(Cy) ~L24(W) ~L25(NT) 12~L(W 2W*,W) (1:34)



~L31(W) ~L32(Cx) ~L33(Cy) ~L34(F) ~L35(NT) ~L36(ST) I5 W,x I3 Cx (1:35)

~L41(W) ~L42(Cx) ~L43(Cy) ~L44(F) ~L45(NT) ~L46(ST) I5 W,y I3 Cy (1:36)

where all linear operators ~Lij() and the nonlinear operator ~L() are dened by

~L11() c1 F11* @4

@X4 F12* F21* 4F66* @

4

@X2@Y2 F22* @

4

@Y4

~L12() D11* c1F11* @3

@X3 D12* 2D66* c1 F12* 2F66* @

3

@X@Y2

~L13() D12* 2D66* c1 F21* 2F66* @3

@X2@Y D22* c1F22* @

3

@Y3

~L14() B21* @4

@X4 B11* B22* 2B66* @

4

@X2@Y2 B12* @

4

@Y4

~L15(NT) @

2

@X2B11* N

Tx B21* N

Ty

2 @

2

@X@YB66* N

Txy

@

2

@Y2B12* N

Tx B22* N

Ty

~L16 M

T

@2

@X2M

Tx

2 @

2

@X@YM

Txy

@

2

@Y2M

Ty

~L17() c1 I5 I4I2I1

@2

@X2 @

2

@Y2

I1

~L21() A22* @4

@X4 2A12* A66* @

4

@X2@Y2 A11* @

4

@Y4

~L22() B21* c1E21* @3

@X3 B11* B66* c1 E11* E66* @

3

@X@Y2

~L23() B22* B66* c1 E22* E66* @3

@X2@Y B12* c1E12* @

3

@Y3

~L24() c1 E21* @4

@X4 E11* E22* 2E66* @

4

@X2@Y2 E12* @

4

@Y4

~L25(NT) @

2

@X2A12* N

Tx A22* N

Ty

@

2

@X@YA66* N

Txy

@

2

@Y2A11* N

Tx A12* N

Ty

~L31() A55 2c2D55 c22F55

@@X

c1 F11* c1H11* @3

@X3

c1 F21* 2F66* c1 H12* 2H66* @3

@X@Y2

~L32() A55 2c2D55 c22F55 D11* 2c1F11* c21H11* @2@X2 D66* 2c1F66* c21H66* @2

@Y2



~L33() D12* D66* c1 F12* F21* 2F66* c21 H12* H66*

@2

@X@Y~L34() ~L22()

~L35(NT) @

@XB11* c1E11* NTx B21* c1E21* N

Ty

h i @

@YB66* c1E66* NTxy

h i~L36(S

T) @

@XSTx

@

@YSTxy

~L41() A44 2c2D44 c22F44

@@Y

c1 F12* 2F66*

c1 H12* 2H66* @3

@X2@Y c1 F22* c1H22* @

3

@Y3~L42() ~L33()~L43() A44 2c2D44 c22F44

D66* 2c1F66* c21H66* @2@X2 D22* 2c1F22* c21H22* @2

@Y2~L44() ~L23()

~L45(NT) @

@XB66* c1E66* NTxy

h i @

@YB12* c1E12* NTx B22* c1E22* N

Ty

h i~L46(S

T) @

@X(S

Txy)

@

@Y(S

Ty )

~L() @2

@X2@2

@Y2 2 @

2

@X@Y@2

@X@Y @

2

@Y2@2

@X2(1:37)

It is worthy to note that the governing differential equations (Equations1.33 through 1.37) for an FGM plate are identical in form to those of unsym-metric cross-ply laminated plates. These general von Krmn-type equationswill be used in solving many nonlinear problems, e.g., nonlinear bending,postbuckling, and nonlinear vibration of shear deformable FGM plates.

References

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Birman V. and Byrd L.W. (2007), Modeling and analysis of functionally gradedmaterials and structures, Applied Mechanics Reviews, 60, 195216.

Chan S.H. (2001), Performance and emissions characteristics of a partially insulatedgasoline engine, International Journal of Thermal Science, 40, 255261.

Fuchiyama T. and Noda N. (1995), Analysis of thermal stress in a plate of functionallygradient material, JSAE Review, 16, 263268.



Gibson L.J., Ashby M.F., Karam G.N., Wegst U., and Shercliff H.R. (1995), Mechanicalproperties of natural materials. II. Microstructures for mechanical efciency,Proceedings of the Royal Society of London Series A, 450, 141162.

Hill R. (1965), A self-consistent mechanics of composite materials, Journal of theMechanics and Physics of Solids, 13, 213222.

Hirai T. and Chen L. (1999), Recent and prospective development of functionallygraded materials in Japan, Materials Science Forum, 308311, 509514.

Koizumi M. (1993), The concept of FGM, Ceramic Transactions, Functionally GradientMaterials, 34, 310.

Librescu L. and Song S.-Y. (2005), Thin-walled beams made of functionally gradedmaterials and operating in a high temperature environment: vibration andstability, Journal of Thermal Stresses, 28, 649712.

Liu L.-S., Zhang Q.-J., and Zhai P.-C. (2003), The optimization design of metal=ceramic FGM armor with neural net and conjugate gradient method, MaterialsScience Forum, 423425, 791796.

Markworth A.J., Ramesh K.S., and ParksW.P. (1995), Review: modeling studies appliedto functionally graded materials, Journal of Material Sciences, 30, 21832193.

Mori T. and Tanaka K. (1973), Average stress in matrix and average elastic energy ofmaterials with mistting inclusions, Acta Metallurgica, 2, 1571574.

Mller E., Draar C., Schilz J., and Kaysser W.A. (2003), Functionally gradedmaterials for sensor and energy applications, Materials Science and Engineering,A362, 1739.

Noda N. (1999), Thermal stresses in functionally graded material, Journal of ThermalStresses, 22, 477512.

Paszkiewicz B., Paszkiewicz R., Wosko M., Radziewicz D., Sciana B., Szyszka A.,Macherzynski W., and Tlaczala M. (2008), Functionally graded semiconductorlayers for devices application, Vacuum, 82, 389394.

Paulino G.H., Jin Z.H., and Dodds Jr. R.H. (2003), Failure of functionally gradedMaterials, in Comprehensive Structural Integrity, Vol. 2 (eds. B. Karihallo andW.G. Knauss), Elsevier Science, New York, pp. 607644.

Qiu J., Tani J., Ueno T., Morita T., Takahashi H., and Du H. (2003), Fabrication andhigh durability of functionally graded piezoelectric bending actuators, SmartMaterials and Structures, 12, 115121.

Reddy J.N. (1984a), A simple high-order theory for laminated composite plates,Journal of Applied Mechanics ASME, 51, 745752.

Reddy J.N. (1984b), A rened nonlinear theory of plates with transverse sheardeformation, International Journal of Solids and Structure, 20, 881896.

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Reuter T. and Dvorak G.J. (1998), Micromechanical models for graded compositeMaterials: II. Thermomechanical loading, Journal of Mechanics and Physics ofSolids, 46, 16551673.

Reuter T., Dvorak G.J., and Tvergaard V. (1997), Micromechanical models for gradedcomposite materials, Journal of Mechanics and Physics of Solids, 45, 12811302.

Shen H.-S. (1997), Krmn-type equations for a higher-order shear deformation platetheory and its use in the thermal postbuckling analysis, Applied Mathematics andMechanics, 18, 11371152.

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2Nonlinear Bending of Shear DeformableFGM Plates

2.1 Introduction

The nonlinear bending response of FGM plates subjected to transversemechanical loads and thermal loadingwas the subject of recent investigations.Previous works for the linear bending of FGM rectangular, circular, andannular plates can be found in Reddy et al. (1999), Cheng and Batra (2000),Reddy and Cheng (2001), Vel and Batra (2002), Croce and Venini (2004),Kashtalyan (2004), and Chung and Chen (2007). Mizuguchi and Ohnabe(1996) employed the Poincare method to examine the large deection ofheated FGM thin plates with Youngs modulus varying symmetrically tothe middle plane in thickness direction. Suresh and Mortensen (1997) pre-sented the large deformation problem of graded multilayered compositesunder thermomechanical loads. When the thermomechanical load reaches ahigh level, nonlinear straindisplacement relations have to be employed. Asa result, a set of nonlinear equations will appear no matter what kind ofanalysis method is used. Based on the FSDPT, Praveen and Reddy (1998)analyzed nonlinear static and dynamic response of functionally gradedceramicmetal plates subjected to transverse mechanical loads and a one-dimensional (1D) steady heat conduction by using nite element method(FEM). This work was then extended to the case of FGM square plates andshallow shell panels by Woo and Meguid (2001) using Fourier series tech-nique, and to the case of FGM circular plates by Ma and Wang (2003) andGunes and Reddy (2008), and to the case of FGM rectangular plates byGhannadPour and Alinia (2006) and Ovesy and GhannadPour (2007) usingRitz method and nite strip method, respectively. However, in their studiesthe formulations were based on the classical plate=shell theory, i.e., the theorybased on the KirchhoffLove hypothesis and therefore the transverse sheardeformations were not accounted for, and the material properties wereassumed to be independent of temperature. Reddy (2000) developed theoret-ical formulations for thick FGM plates according to the HSDPT. In his study,both Navier solutions for linear bending of simply supported rectangularFGM plates and nite element models for nonlinear static and dynamicresponse were presented. The paper of Cheng (2001) also contains the

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C002 Final Proof page 21 8.12.2008 9:38am Compositor Name: DeShanthi

21

solution for nonlinear bending of transversely isotropic symmetric sheardeformable FGM plates. Moreover, Shen (2002) provided a nonlinear bend-ing analysis of simply supported shear deformable FGM rectangular platessubjected to a transverse uniform or sinusoidal load and in thermal environ-ments. In his study, the material properties were considered to be tempera-ture dependent and the effect of temperature rise on the nonlinear bendingresponse was reported. Subsequently, Yang and Shen (2003a,b) developed asemianalytical-numerical method to examine the large deection of thin andshear deformable FGM rectangular plates subjected to combined mechanicaland thermal loads and under various boundary conditions. This method wasthen extended to the case of FGM hybrid plates with surface-bonded piezo-electric layers by Yang et al. (2004). In these studies, the material propertieswere assumed to be temperature independent and temperature dependent,respectively. Recently, Na and Kim (2006) studied nonlinear bending ofclamped FGM rectangular plates subjected to a transverse uniform pressureand thermal loads by using a 3D FEM. In their study, the thermal loads wereassumed as uniform, linear, and sinusoidal temperature rises across thethickness direction, whereas the material properties were assumed to betemperature independent. On the other hand, ceramics and the metals usedin FGMs do store different amounts of heat, and therefore the heat conduc-tion usually occurs (Tanigawa et al. 1996, Kim and Noda 2002). This leads toa nonuniform distribution of temperature through the plate thickness, but itis not accounted for in the above studies. This is because when the materialproperties are assumed to be functions of temperature and position, and thetemperature is also assumed to be a function of position, the problembecomes very complicated. More recently, Shen (2007) provided a nonlinearthermal bending analysis of simply supported shear deformable FGM rect-angular plates due to heat conduction. In his study, both heat conduction andtemperature-dependent material properties were taken into account.

2.2 Nonlinear Bending of FGM Plates under MechanicalLoads in Thermal Environments

Here, we consider an FGM plate of length a, width b, and thickness h, whichis made from a mixture of ceramics and metals. We assume that the com-position is varied from the top to the bottom surface, i.e., the top surface(Zh=2) of the plate is ceramic-rich whereas the bottom surface (Z h=2) ismetal-rich. The plate is subjected to a transverse uniform load q q0 or asinusoidal load q q0 sin(pX=a)sin(pY=b) combined with thermal loads.It is assumed that Ec, Em, ac, and am are functions of temperature, but

Poissons ratio nf depends weakly on temperature change and is assumed tobe a constant. We assume the volume fraction Vm follows a simple powerlaw as expressed by Equation 1.1. According to mixture rules, the effective



Youngs modulus Ef and thermal expansion coefcient af of an FGM platecan be written as


Ec(T) (2:1a)


ac(T) (2:1b)

It is evident that when Zh=2, EfEc and afac, and when Z h=2,EfEm and afam.In the case of a transverse static load applied at the top surface of an FGM

plate, the general von Krmn-type equations (Equations 1.33 through 1.36)can be written in the simple form as

~L11(W) ~L12(Cx) ~L13(Cy) ~L14(F) ~L15(NT) ~L16(MT) ~L(W,F) q (2:2)

~L21(F) ~L22(Cx) ~L23(Cy) ~L24(W) ~L25(NT) 12~L(W,W) (2:3)

~L31(W) ~L32(Cx) ~L33(Cy) ~L34(F) ~L35(NT) ~L36(ST) 0 (2:4)~L41(W) ~L42(Cx) ~L43(Cy) ~L44(F) ~L45(NT) ~L46(ST) 0 (2:5)

Note that the geometric nonlinearity in the von Krmn sense is given interms of ~L() in Equations 2.2 and 2.3, and the other linear operators ~Lij() aredened by Equation 1.37, and the forces, moments, and higher ordermoments caused by elevated temperature are dened by Equation 1.28.All the edges are assumed to be simply supported. Depending upon the

in-plane behavior at the edges, two cases, case 1 (referred to herein as movableedges) and case 2 (referred to herein as immovable edges), will be considered.

Case 1. The edges are simply supported and freely movable in both theX- and Y-directions, respectively.

Case 2. All four edges are simply supported with no in-plane displacements,i.e., prevented from moving in the X- and Y-directions.For these two cases the associated boundary conditions areX 0, a:

W Cy 0 (2:6a)b0

NxdY 0 (movable edges) (2:6b)

U 0 (immovable edges) (2:6c)


Nonlinear Bending of Shear Deformable FGM Plates 23

Y 0, b:W Cx 0 (2:6d)

a0

NydX 0 (movable edges) (2:6e)

V 0 (immovable edges) (2:6f)

It is noted that the presence of stretchingbending coupling gives rise tobending curvatures under the action of in-plane loading, no matter howsmall these loads may be. In this situation, the boundary condition of zerobending moment cannot be incorporated accurately. Because immovableedges are considered in the present analysis, Mx 0 (at X 0, a) and My 0(at Y 0, b) are not included in Equation 2.6, as previously shown inKuppusamy and Reddy (1984) and Singh et al. (1994).The unit end-shortening relationships are

Dxa 1

ab

b0

a0

@U@X

dX dY

1ab

b0

a0

("A11*

@2F@Y2

A12* @2F

@X2 B11* 43h2 E11*

@Cx@X

B12* 43h2 E12*

@Cy

@Y 43h2

E11*@2W@X2

E12* @2W@Y2

#

12

@W@X

2 A11* NTx A12* N

Ty

)dX dY (2:7a)

Dy

b 1

ab

a0

b0

@V@Y

dYdX

1ab

a0

b0

("A22*

@2F@X2

A12* @2F

@Y2 B21* 43h2 E21*

@Cx@X

B22* 43h2 E22*

@Cy

@Y 43h2

E21*@2W@X2

E22* @2W@Y2

#

12

@W@Y

2 A12* NTx A22* N

Ty

)dYdX (2:7b)



where Dx and Dy are plate end-shortening displacements in the X- andY-directions.Having developed the theory, we are now in a position to solve Equa-

tions 2.2 through 2.5 with boundary condition (Equation 2.6). Before pro-ceeding, it is convenient to rst dene the following dimensionless quantitiesfor such plates (with gijk in Equations 2.14 and 2.16 below are dened as inAppendix A):

x pX=a, y pY=b, b a=b, W W=[D11* D22* A11* A22* ]1=4

F F=[D11* D22* ]1=2, (Cx,Cy) (Cx,Cy)a=p[D11* D22* A11* A22* ]1=4

g14 [D22* =D11* ]1=2, g24 [A11* =A22* ]1=2, g5 A12* =A22*

(gT1,gT2) (ATx ,ATy )a2=p2[D11* D22* ]1=2,

(gT3,gT4,gT6,gT7) (DTx ,DTy ,FTx ,FTy )a2=p2h2D11* ,(Mx,My,Px,Py,MTx ,M

Ty ,P

Tx ,P

Ty )

(Mx,My,4Px=3h2,4Py=3h2,MTx ,MTy ,4P

Tx=3h

2,4PTy=3h

2)a2=p2D11* [D11* D22* A11* A22* ]1=4

lq q0a4=p4D11* [D11* D22* A11* A22* ]1=4,(dx,dy) (Dx=a,Dy=b)b2=4p2[D11* D22* A11* A22* ]1=2 (2:8)

where ATx (ATy ), DTx (DTy ), and FTx (FTy ) are dened by

ATx DTx F

Tx

ATy DTy F

Ty

" #

h=2h=2

Ax

Ay

" #(1,Z,Z3)dZ (2:9)

and the details of which can be found in Appendix B.The nonlinear governing equations (Equations 2.2 through 2.5) can then be

written in dimensionless form as

L11(W) L12(Cx) L13(Cy) g14L14(F) L16(MT) g14b2L(W,F) lq (2:10)

L21(F) g24L22(Cx) g24L23(Cy) g24L24(W) 12g24b

2L(W,W) (2:11)

L31(W) L32(Cx) L33(Cy) g14L34(F) L36(ST) 0 (2:12)

L41(W) L42(Cx) L43(Cy) g14L44(F) L46(ST) 0 (2:13)



where

L11() g110@4

@x4 2g112b2

@4

@x2@y2 g114b4

@4

@y4

L12() g120@3

@x3 g122b2

@3

@x@y2

L13() g131b@3

@x2@y g133b3

@3

@y3

L14() g140@4

@x4 g142b2

@4

@x2@y2 g144b4

@4

@y4

L16(MT) @2

@x2(MTx ) 2b

@2

@x@y(MTxy) b2

@2

@y2(MTy )

L21() @4

@x4 2g212b2

@4

@x2@y2 g214b4

@4

@y4

L22() g220@3

@x3 g222b2

@3

@x@y2

L23() g231b@3

@x2@y g233b3

@3

@y3

L24() g240@4

@x4 g242b2

@4

@x2@y2 g244b4

@4

@y4

L31() g31@

@x g310

@3

@x3 g312b2

@3

@x@y2

L32() g31 g320@2

@x2 g322b2

@2

@y2

L33() g331b@2

@x@yL34() L22()

L36(ST) @@x

(STx ) b@

@y(STxy)

L41() g41b@

@y g411b

@3

@x2@y g413b3

@3

@y3

L42() L33()

L43() g41 g430@2

@x2 g432b2

@2

@y2

L44() L23()L46(ST) @

@x(STxy) b

@

@y(STy )

L() @2

@x2@2

@y2 2 @

2

@x@y@2

@x@y @

2

@y2@2

@x2

(2:14)



The boundary conditions of Equation 2.6 become

x 0, p:W Cy 0 (2:15a)

p0

b2@2F@y2

dy 0 (movable edges) (2:15b)

dx 0 (immovable edges) (2:15c)y 0, p:

W Cx 0 (2:15d)p0

@2F@x2

dx 0 (movable edges) (2:15e)

dy 0 (immovable edges) (2:15f)

and the unit end-shortening relationships become

dx 14p2b2g24

p0

p0

g224b2@

2F@y2

g5@2F@x2

g24 g511@Cx@x

g233b@Cy

@y

g24 g611@2W@x2

g244b2@2W@y2

12g24

@W@x

2(g224gT1g5gT2)DT

)dxdy (2:16a)

dy 14p2b2g24

p0

p0

@2F@x2

g5b2@2F@y2

g24 g220@Cx@x

g522b@Cy

@y

g24 g240@2W@x2

g622b2@2W@y2

12g24b

2 @W@y

2(gT2g5gT1)DT

)dydx (2:16b)

Applying Equations 2.10 through 2.16, the nonlinear bending response of asimply supported FGM plate subjected to a transverse uniform or sinusoidalload and in thermal environments is now determined by means of a two-stepperturbation technique, for which the small perturbation parameter has nophysical meaning at the rst step, and is then replaced by a dimensionlesscentral deection at the second step. The essence of this procedure, in thepresent case, is to assume that

W(x, y, ) Xj1

jwj(x, y), F(x, y, ) Xj0

jfj(x, y)

Cx(x, y, ) Xj1

jcxj(x, y), Cy(x, y, ) Xj1

jcyj(x, y), lq Xj1

jlj

(2:17)



where is a small perturbation parameter and the rst term of wj(x, y) isassumed to have the form:

w1(x, y) A(1)11 sinmx sin ny (2:18)

Then, we expand the thermal bending moments in the double Fourier sineseries as

MTx STx

MTy STy

" # M

(1)x S

(1)x

M(1)y S(1)y

" # Xi1,3,...

Xj1,3,...

1ijsin ix sin jy (2:19)

where M(1)x , M(1)y , S

(1)x , and S

(1)y and all coefcients in Equations 2.32 through

2.34 below are given in detail in Appendix C.Substituting Equation 2.17 into Equations 2.10 through 2.13, collecting the

terms of the same order of , we obtain a set of perturbation equations whichcan be written, for example, as

O(0):

L14( f0) 0 (2:20a)L21( f0) 0 (2:20b)L34( f0) 0 (2:20c)L44( f0) 0 (2:20d)

O(1):

L11(w1) L12(cx1) L13(cy1) g14L14( f1) g14b2L(w1, f0) l1 (2:21a)L21( f1) g24L22(cx1) g24L23(cy1) g24L24(w1) 0 (2:21b)

L31(w1) L32(cx1) L33(cy1) g14L34( f1) 0 (2:21c)L41(w1) L42(cx1) L43(cy1) g14L44( f1) 0 (2:21d)

O(2):

L11(w2) L12(cx2) L13(cy2) g14L14( f2) g14b2[L(w2, f0) L(w1, f1)] l2(2:22a)

L21( f2) g24L22(cx2) g24L23(cy2) g24L24(w2) 12g24b

2L(w1,w1) (2:22b)




O(3):

L11(w3) L12(cx3) L13(cy3) g14L14( f3) g14b2[L(w3, f0) L(w2, f1) L(w1, f2)] l3 (2:23a)

L21( f3) g24L22(cx3) g24L23(cy3) g24L24(w3) 12g24b

2L(w1,w2) (2:23b)


To solve these perturbation equations of each order, the amplitudes of theterms wj(x, y), fj(x, y), cxj(x, y), and cyj(x, y) can be determined step by step,and lj can be determined by the Galerkin procedure. As a result, up to third-order asymptotic solutions can be obtained as

W A(1)11 sinmx sinnyh i

3 A(3)13 sinmx sin 3nyA(3)31 sin 3mx sinnyh i

O(4)(2:24)

Cx C(1)11 cosmx sin nyh i

2 C(2)20 sin 2mxh i

3 C(3)13 cosmx sin 3ny C(3)31 cos 3mx sin nyh i

O(4) (2:25)

Cy D(1)11 sinmx cos nyh i

2 D(2)02 sin 2nyh i

3 D(3)13 sinmx cos 3nyD(3)31 sin 3mx cos nyh i

O(4) (2:26)

F B(0)00y2

2 b(0)00

x2

2 B(1)11 sinmx sin ny

h i 2 B(2)00

y2

2 b(2)00

x2

2 B(2)20 cos 2mx B(2)02 cos 2ny

3 B(3)13 sinmx sin 3ny B(3)31 sin 3mx sin nyh i

O(4) (2:27)

Note that for boundary condition case 1, it is just necessary to takeB(i)00 b(i)00 0 (i 0, 2) in Equation 2.27, so that the asymptotic solutionshave a similar form, and

lq l1 2l2 3l3 O(4) (2:28)

All coefcients in Equations 2.24 through 2.27 are related and can be writtenas functions of A(1)11 , for example



B(1)11 g24g05g06

A(1)11 , B(2)20

g24n2b2

32m2g6A(1)11 2

, B(2)02 g24m

2

32n2b2g7A(1)11 2

C(1)11 mg04g00

g14g24g02g05g00g06

A(1)11 , C

(2)20 g14g24g220

mn2b2

4 g31g3204m2 g6A(1)11 2

D(1)11 nbg03g00

g14g24g01g05g00g06

A(1)11 , D

(2)02 g14g24g233

m2nb4 g41g4324n2b2

g7A(1)11 2(2:29)

All symbols used in Equation 2.29 are also described in detail in Appendix C.Hence Equations 2.24 and 2.28 can be rewritten as

W W(1)(x, y) A(1)11

W(3)(x,y) A(1)11 3

(2:30)

and

lq l(1)q A(1)11

l(2)q A(1)11 2

l(3)q A(1)11 3

(2:31)

From Equations 2.30 and 2.31 the loadcentral deection relationship can bewritten as

q0a4

D11* h A(0)W A(1)W

Wh

A(2)W

Wh

2A(3)W

Wh

3 (2:32)

Similarly, the bending momentcentral deection relationships can bewritten as

Mxa2

D11* h A(0)MX A(1)MX

Wh

A(2)MX

Wh

2A(3)MX

Wh

3 (2:33)

Mya2

D11* h A(0)MY A(1)MY

Wh

A(2)MY

Wh

2A(3)MY

Wh

3 (2:34)

Equations 2.32 through 2.34 can be employed to obtain numerical resultsfor the loaddeection and loadbending moment curves of an FGM platesubjected to a transverse uniform or sinusoidal load and in thermal environ-ments. Zirconia and titanium alloys were selected for the two constituentmaterials of the plate in the present examples, referred to as ZrO2=Ti-6Al-4V.The material properties of these two constituents are assumed to be nonlinearfunction of temperature of Equation 1.4 (Touloukian 1967), and typicalvalues for Youngs modulus Ef (in Pa) and thermal expansion coefcient af



(in K1) of Zirconia and Ti-6Al-4V can be found in Tables 1.1 and 1.2 (fromReddy and Chin 1998). Poissons ratio nf is assumed to be a constant, andnf 0.28. The results presented herein are for movable in-plane boundaryconditions, unless it is stated otherwise.The loadcenter deection curves for a zirconia=aluminum square plate

with different values of the volume fraction index subjected to a uniformtransverse load are compared in Figure 2.1 with numerical results of Praveenand Reddy (1998), using their material properties, i.e., for aluminum E 70GPa, n 0.3, a 23.0 106 8C1, and for zirconia E 151 GPa, n 0.3, anda 10.0 106 8C1. Note that in Figure 2.1 the volume fraction index N isdened for Vc, and E0 is a referenced value of Youngs modulus, and E0 70GPa. Clearly, the comparison is reasonably well.Figure 2.2 gives the loaddeection and loadbending moment curves of

ZrO2=Ti-6Al-4V square plate with different values of volume fraction indexN ( 0, 0.5, 1.0, 2.0, 5.0, and 1) subjected to a uniform pressure and underthermal environmental condition DT 0 K. The results show that a fullytitanium alloy plate (N 0) has highest deection and lowest bendingmoment. It can also be seen that the plate has higher deection and lowerbending moment when it has lower volume fraction. This is expected becausethe metallic plate is the one with the lower stiffness than the ceramic plate.Figure 2.3 gives the loaddeection and loadbending moment curves of

ZrO2=Ti-6Al-4V square plates subjected to a uniform pressure and underthree sets of thermal environmental conditions, referred to as I, II, and III. Forenvironmental condition I, DT 0 K; for environmental condition II,

0.0

0.2

0.4

0.6

PresentPraveen and Reddy (1998)

1: Aluminum2: N = 2.03: N = 1.04: N = 0.55: Zirconia

5

4

3

2

1

Uniform loadZirconia/aluminumb =1.0, b/h = 20

q0b4/E0h

4

W/h

0 5 10 15

FIGURE 2.1Comparisons of loadcentral deection curves for a zirconia=aluminum square plate.



DT 200 K; and for environmental condition III, DT 300 K. Because thethermal expansion at the top surface is higher than that at the bottom surface,this expansion results in an upward deection. It is seen that the deectionsare reduced, but the bending moments are increased with increases intemperature. Note that for environmental conditions II and III the deectionsare close to each other when W=h> 1.5.

00.0

0.5

1.0

1.5

2.0W

/h

ZrO2N = 5.0N = 2.0N = 1.0N = 0.5Ti-6Al-4V

ZrO2/Ti-6Al-4Vb = 1.0, b/h = 20

Uniform loadT0 = 300 K

(a)

50 100 150

q0b4/E0h

4

0

1

2

3

(b)

ZrO2N = 5.0N = 2.0N = 1.0N = 0.5Ti-6Al-4V

ZrO2/Ti-6Al-4Vb =1.0, b/h = 20


0 50 100 150q0b

4/E0h4

Mxb

2 /E0h

4

FIGURE 2.2Effect of volume fraction index N on the nonlinear bending behavior of ZrO2=Ti-6Al-4V squareplates under uniform pressure: (a) loadcentral deection; (b) loadbending moment.



Figure 2.4 shows the effect of in-plane boundary conditions on the non-linear bending behavior of ZrO2=Ti-6Al-4V square plates under two envir-onmental conditions. To this end, the loaddeection and loadbendingmoment curves of ZrO2=Ti-6Al-4V square plates under movable andimmovable in-plane boundary conditions are displayed. The results

00.0

0.5

1.0

1.5

2.0W

/h

N = 5.0 (III)N = 0.5 (III)N = 5.0 (II)N = 0.5 (II)N = 5.0 (I)N = 0.5 (I)

ZrO2/Ti-6Al-4Vb =1.0, b/h = 20


(a)

I: T = 0 KII: T = 200 KIII: T = 300 K

50 100 150q0b

4/E0h4

0 50 100 1500

1

2

3

(b)

N = 5.0 (III)N = 0.5 (III)N = 5.0 (II)N = 0.5 (II)N = 5.0 (I)N = 0.5 (I)

ZrO2/Ti-6Al-4Vb = 1.0, b/h = 20


I: T = 0 KII: T = 200 KIII: T = 300 K

q0b4/E0h

4

Mxb

2 /E0h

4

FIGURE 2.3Effect of temperature rise on the nonlinear bending behavior of ZrO2=Ti-6Al-4V square platesunder uniform pressure: (a) loadcentral deection; (b) loadbending moment.



show that the plate with immovable edges will undergo less deection withsmaller bending moments.Figure 2.5 compares the loaddeection and loadbending moment curves

of ZrO2=Ti-6Al-4V square plates under two cases of transverse loadingconditions along with two environmental conditions. It can be seen that both

00

1

2

31: Movable edges2: Immovable edges

ZrO2/Ti-6Al-4Vb =1.0, b/h = 20

q0b4/E0h

4

I: T = 0 KII: T = 200 K


W/h

N = 5.0 (II & 2)N = 0.5 (II & 2)N = 5.0 (I & 2)N = 0.5 (I & 2)N = 5.0 (II & 1)N = 0.5 (II & 1)N = 5.0 (I & 1)N = 0.5 (I & 1)

(a)

50 100 150

00

1

2

3

4N = 5.0 (II & 2)N = 0.5 (II & 2)N = 5.0 (I & 2)N = 0.5 (I & 2)N = 5.0 (II & 1)N = 0.5 (II & 1)N = 5.0 (I & 1)N = 0.5 (I & 1)

I: T = 0 KII: T = 200 K

q0b4/E0h

4

Mxb

2 /E 0

h4

1: Movable edges2: Immovable edges

ZrO2/Ti-6Al-4Vb = 1.0, b/h = 20


(b)50 100 150

FIGURE 2.4Effect of in-plane boundary conditions on the nonlinear bending behavior of ZrO2=Ti-6Al-4Vsquare plates under uniform pressure: (a) loadcentral deection; (b) loadbending moment.



loaddeection and loadbending moment curves of the plate subjected to asinusoidal load are lower than those of the plate subjected to a uniform load.It is appreciated that in Figures 2.2 through 2.5, W=h, Mxb

2=E0h4, and

q0b4=E0h

4 denote the dimensionless central deection of the plate, centralbending moment, and lateral pressure, respectively, where E0Youngsmodulus of Ti-6Al-4V at T 300 K.

00.0

0.5

1.0

1.5

2.0

1: Uniform load2: Sinusoidal load

ZrO2/Ti-6Al-4Vb = 1.0, b/h = 20

q0b4/E0h

4

I: T = 0 KII: T = 200 K

T0 = 300 K

W/h


(a) 50 100 150

0.0

0.5

1.0

1.5

2.0

2.5

1: Uniform load2: Sinusoidal load


Mxb

2 /E 0

h4

(b)

I: T = 0 KII: T = 200 K

ZrO2/Ti-6Al-4Vb = 1.0, b/h = 20T0 = 300 K

0q0b

4/E0h4

50 100 150

FIGURE 2.5Comparisons of nonlinear responses of ZrO2=Ti-6Al-4V square plates subjected to a uniform orsinusoidal load and in thermal environments: (a) loadcentral deection; (b) loadbendingmoment.



2.3 Nonlinear Thermal Bending of FGM Platesdue to Heat Conduction

In the case of a temperature eld applied at the top and bottom surfaces of anFGM plate, thermal bending usually occurs due to heat conduction. In thepresent case, we assume that the effective Youngs modulus Ef, thermalexpansion coefcient af, and thermal conductivity kf are functions of tem-perature, so that Ef, af, and kf are both temperature- and position-dependent.The Poisson ratio nf depends weakly on temperature change and is assumedto be a constant. According to mixture rules, we have


Ec(T) (2:35a)


ac(T) (2:35b)

kf(Z,T) [km(T) kc(T)] 2Z h2h N

kc(T) (2:35c)

We assume that the temperature variation occurs in the thickness directiononly and 1D temperature eld is assumed to be constant in the XY plane ofthe plate. In such a case, the temperature distribution along the thickness canbe obtained by solving a steady-state heat transfer equation:

ddZ

kdTdZ

0 (2:36)

Equation 2.36 is solved by imposing the boundary conditions TTU atZh=2 and TTL at Z h=2. The solution of this equation, by means ofpolynomial series, is (Javaheri and Eslami 2002):

T(Z) TU (TL TU)h(Z) (2:37)

where TU and TL are the temperatures at top and bottom surfaces of theplate, and

h(Z) 1C

2Z h2h

kmc(N 1)kc

2Z h2h

N1 k

2mc

(2N 1)k2c2Z h2h

2N1"

k3mc

(3N 1)k3c2Z h2h

3N1 k

4mc

(4N 1)k4c2Z h2h

4N1

k5mc

(5N 1)k5c2Z h2h

5N1#(2:38a)



C 1 kmc(N 1)kc

k2mc(2N 1)k2c

k3mc

(3N 1)k3c k

4mc

(4N 1)k4c k

5mc

(5N 1)k5c(2:38b)

where kmc km kc. In particular, for an isotropic material, Equation 2.37may then be expressed as

T(Z) TU TL2

TU TLh

Z (2:39)

In the present case, since there is no transverse static load applied, thenonlinear governing equations can be written in dimensionless form as

L11(W) L12(Cx) L13(Cy) g14L14(F) g14b2L(W,F) L16(MT) (2:40)

L21(F) g24L22(Cx) g24L23(Cy) g24L24(W) 12g24b

2L(W,W) (2:41)

L31(W) L32(Cx) L33(Cy) g14L34(F) L36(ST) (2:42)

L41(W) L42(Cx) L43(Cy) g14L44(F) L46(ST) (2:43)

where all nondimensional linear operators Lij() and nonlinear operator L() aredened as expressed by Equation 2.14. Note that Equation 2.9 is nowredened by

ATx DTx F

Tx

ATy DTy F

Ty

" #DT

h=2h=2

AxAy

" #(1,Z,Z3)DT(X,Y,Z)dZ (2:44)

where DT is a constant and is dened by DTTUTL. WhenDT 0, ATx (ATy ), DTx (DTy ), and FTx (FTy ) can be found in Appendix B,and if DT 6 0, then ATx , DTx , and FTx can be found in Appendix D.L16(M

p), L36(Sp), and L46(S

p) in Equations 2.40, 2.42, and 2.43 are nowtreated as pseudoloads, and we expand the thermal bending moments inthe double Fourier sine series as

MTx STx

MTy STy

" # M

0x S

0x

M0y S0y

" # Xi1,3,...

Xj1,3,...

1ijsin ix sin jy (2:45)

where M0x, M0y, S

0x, and S

0y are given in detail in Appendix E.



By using a two-step perturbation technique, we obtain up to third-orderasymptotic solutions in the same form as expressed by Equations 2.24through 2.27, from which we have

Wt A(1)W A(1)11

A(3)W A(1)11

3 (2:46)

and the bending moments can be written as

Mxa2

D11* t A(0)MX A(1)MX A(1)11

A(2)MX A(1)11

2A(3)MX A(1)11

3 (2:47)

Mya2

D11* t A(0)MY A(1)MY A(1)11

A(2)MY A(1)11

2A(3)MY A(1)11

3 (2:48)

in Equations 2.46 through 2.48, A(2)11

is taken as the second perturbationparameter relating to the temperature variation, i.e.,

A(2)11 lQ2(l)2 Q3(l)3 (2:49)

All symbols used in Equations 2.46 through 2.49 are also described in detailin Appendix E.Equations 2.46 through 2.49 can be employed to obtain numerical results

for the thermal loaddeection and thermal loadbending moment curves ofan FGM plate under heat conduction. Silicon nitride and stainless steel wereselected for the two constituent materials of the substrate FGM layer, referredto as Si3N4=SUS304, in the present examples. The material properties of thesetwo constituents are assumed to be nonlinear function of temperature ofEquation 1.4, and typical values for Youngs modulus Ef (in Pa), thermalexpansion coefcient af (in K

1), and the thermal conductivity kf (inW mK1) of silicon nitride and stainless steel can be found in Tables 1.1through 1.3. Poissons ratio nf is assumed to be a constant, and nf 0.28.For these examples, the lower surface is hold at a prescribed temperature of300 K, so that DTL 0 and DTDTU.Figure 2.6 presents the thermal loaddeection and thermal loadbending

moment curves for square FGM plates with different values of volumefraction index N ( 0, 0.2, 0.5, 1.0, 2.0, 5.0, and 1) under temperaturevariation DTU at upper surface. It can be seen that the deection of FGMplates with lower values of volume fraction index N is positive (downward),whereas for the plate with higher values of N the deection becomes nega-tive. This is due to the fact that the thermal expansion coefcient at the lowersurface is larger than that experienced by the upper surface. The results showthat the plate has higher bending moment (except for N 0 and N 0.2)



when it has lower volume fraction. It can also be seen that the bendingmoment for a fully stainless steel plate (N 0) changes from positive tonegative when the temperature variation DTU> 150 K. The results revealthat the nonlinear bending responses of an FGM plate due to heat conduc-tion are quite different to those of an FGM plate subjected to transversemechanical loads.

01.0

0.5

0.0

0.5

1.0

N = 0.5N = 0.2SUS304

Si3N4/SUS304

T0 = TL = 300 Kb = 1.0, b/h = 20

W (m

m)

Si3N4N = 5.0N = 2.0N = 1.0

TU (K)(a)200 400 600

0(b)

100

50

0

50

100

SUS304

Si3N4N = 5.0

N = 0.5N = 0.2N = 2.0

N = 1.0

Mx (

N m

/m)

TU (K)

Si3N4/SUS304b = 1.0, b/h = 20 T0 = TL = 300 K

200 400 600

FIGURE 2.6Effect of volume fraction index N on the nonlinear bending behavior of Si3N4=SUS304 squareplates due to heat conduction: (a) loadcentral deection; (b) loadbending moment.



Figure 2.7 presents the thermal loaddeection and thermal loadbendingmoment curves for square FGM plates with two values of volume fractionindex N 0.5 and 2.0 under two cases of thermoelastic properties TD andTID. Here, TD and TID represent, respectively, the material properties aretemperature dependent and temperature independent, i.e., in a xed tem-perature T0 300 K. Great differences could be seen in these two cases and

00.25

0.00

0.25

0.50

0.75II & N = 2.0II & N = 0.5I & N = 2.0 I & N = 0.5

I: TDII: TID

Si3N4/SUS304b =1.0, b/h = 20T0 = TL = 300 K

TU (K)(a)200 400 600

W (m

m)

025

0

25

50

75

100

Si3N4/SUS304b =1.0, b/h = 20T0 = TL = 300 K

I: TDII: TID

Mx (

N m

)

TU (K)(b)

200 400 600

II & N = 2.0II & N = 0.5I & N = 2.0 I & N = 0.5

FIGURE 2.7Effect of temperature-dependency on the nonlinear bending behavior of Si3N4=SUS304 squareplates due to heat conduction: (a) loadcentral deection; (b) loadbending moment.



we believe that the temperature dependency of FGMs could not be neglectedin the thermal bending analysis.Figure 2.8 shows the effect of in-plane boundary conditions on the non-

linear thermal bending behavior of FGM plates with N 0.5 and 2.0.The thermal loaddeection and thermal loadbending moment curves ofsquare FGM plates under movable and immovable in-plane boundary

00.50

0.25

0.00

0.25

0.50

Si3N4/SUS304b =1.0, b/h = 20T0 = TL = 300 K

I: Immovable edgesII: Movable edges

II & N = 2.0II & N = 0.5I & N = 2.0 I & N = 0.5

W (m

m)

TU (K)(a)200 400 600

0100

50

0

50

100

I: Immovable edgesII: Movable edges

Mx (

N m

)

Si3N4/SUS304b = 1.0, b/h = 20

(b)200 400 600

II & N = 2.0II & N = 0.5I & N = 2.0 I & N = 0.5

TU (K)

T0 = TL = 300 K

FIGURE 2.8Effect of in-plane boundary conditions on the nonlinear bending behavior of Si3N4=SUS304plates due to heat conduction: (a) loadcentral deection; (b) loadbending moment.



conditions are displayed. The results show that the plate with immovableedges will undergo less deection with larger bending moments. It can alsobe seen that for the plate with movable edges both deection and bendingmoment are decreased by increasing the volume fraction index N.

References

Cheng

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Functionally Graded Materials