Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

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Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C000 Final Proof page i 10.12.2008 10:36am Compositor Name: MSubramanian

Functionally Graded Materials Nonlinear Analysis of Plates and Shells

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C000 Final Proof page ii 10.12.2008 10:36am Compositor Name: MSubramanian

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Functionally Graded Materials Nonlinear Analysis of Plates and Shells

Hui-Shen Shen

Boca Raton London New York

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

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-9256-1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. 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 2009 624.1’776--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2008036386

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Contents Preface................................................................................................................... vii Author..................................................................................................................... ix

Chapter 1

Modeling of Functionally Graded Materials and Structures ....................................................................... 1 1.1 Introduction ................................................................................................... 1 1.2 Effective Material Properties of FGMs ....................................................... 3 1.3 Reddy’s Higher Order Shear Deformation Plate Theory........................ 9 1.4 Generalized Kármán-Type Nonlinear Equations ................................... 14 References.............................................................................................................. 17 Chapter 2 Nonlinear Bending of Shear Deformable FGM Plates ... 21 2.1 Introduction ................................................................................................. 21 2.2 Nonlinear Bending of FGM Plates under Mechanical Loads in Thermal Environments .............................................................. 22 2.3 Nonlinear Thermal Bending of FGM Plates due to Heat Conduction............................................................................. 36 References.............................................................................................................. 42 Chapter 3 Postbuckling of Shear Deformable FGM Plates ............. 45 3.1 Introduction ................................................................................................. 45 3.2 Postbuckling of FGM Plates with Piezoelectric Actuators under Thermoelectromechanical Loads................................................... 47 3.3 Thermal Postbuckling Behavior of FGM Plates with Piezoelectric Actuators ...................................................................... 66 3.4 Postbuckling of Sandwich Plates with FGM Face Sheets in Thermal Environments .......................................................................... 82 References.............................................................................................................. 96 Chapter 4

Nonlinear Vibration of Shear Deformable FGM Plates .......................................................................... 99 4.1 Introduction ................................................................................................. 99 4.2 Nonlinear Vibration of FGM Plates in Thermal Environments ......... 100 4.2.1 Free Vibration ................................................................................ 108 4.2.2 Forced Vibration............................................................................ 108 4.3 Nonlinear Vibration of FGM Plates with Piezoelectric Actuators in Thermal Environments ........................................................................ 118 4.4 Vibration of Postbuckled Sandwich Plates with FGM Face Sheets in Thermal Environments ................................................... 132 References............................................................................................................ 143 v

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Contents

Chapter 5 Postbuckling of Shear Deformable FGM Shells ........... 147 5.1 Introduction ............................................................................................... 147 5.2 Boundary Layer Theory for the Buckling of FGM Cylindrical Shells....................................................................................... 149 5.2.1 Donnell Theory.............................................................................. 149 5.2.2 Generalized Kármán–Donnell-Type Nonlinear Equations ..... 150 5.2.3 Boundary Layer-Type Equations ................................................ 152 5.3 Postbuckling Behavior of FGM Cylindrical Shells under Axial Compression ........................................................................ 153 5.4 Postbuckling Behavior of FGM Cylindrical Shells under External Pressure ........................................................................... 169 5.5 Postbuckling Behavior of FGM Cylindrical Shells under Torsion ............................................................................................ 183 5.6 Thermal Postbuckling Behavior of FGM Cylindrical 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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Preface With the development of new industries and modern processes, many structures serve in thermal environments, resulting in a new class of composite materials called functionally graded materials (FGMs). FGMs were initially designed as thermal barrier materials for aerospace structural applications and fusion reactors. They are now developed for general use as structural components in extremely high-temperature environments. The ability to predict the response of FGM plates and shells when subjected to thermal and mechanical loads is of prime interest to structural analysis. In fact, many structures are subjected to high levels of load that may result in nonlinear load–deﬂection relationships due to large deformations. One of the important problems deserving special attention is the study of their nonlinear response to large deﬂection, postbuckling, and nonlinear vibration. This book consists of ﬁve chapters. The chapter and section titles are signiﬁcant indicators of the content matter. Each chapter contains adequate introductory material to enable engineering graduates who are familiar with the basic understanding of plates and shells to follow the text. The modeling of FGMs and structures is introduced and the derivation of the governing equations of FGM plates in the von Kármán sense is presented in Chapter 1. In Chapter 2, the geometrically nonlinear bending of FGM plates due to transverse static loads or heat conduction is presented. Chapter 3 furnishes a detailed treatment of the postbuckling problems of FGM plates subjected to thermal, electrical, and mechanical loads. Chapter 4 deals with the nonlinear vibration of FGM plates with or without piezoelectric actuators. Finally, Chapter 5 presents postbuckling solutions for FGM cylindrical shells under various loading conditions. Most of the solutions presented in these chapters are the results of investigations conducted by the author and his collaborators since 2001. The results presented herein may be treated as a benchmark for 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 the geometrically nonlinear problems of inhomogeneous isotropic and functionally graded plates and shells. It is hoped that this book will ﬁll the gap to some extent and be used as a valuable reference source for postgraduate students, engineers, scientists, and applied mathematicians in this ﬁeld. I wish to record my 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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Author Hui-Shen Shen is a professor of applied mechanics at Shanghai Jiao Tong University. He graduated from Tsinghua University in 1970, and received his MSc in solid mechanics and his PhD in structural mechanics from Shanghai Jiao Tong University in 1982 and 1986, respectively. In 1991–1992, he was a visiting research fellow at the University of Wales (Cardiff) and the University of Liverpool in the United Kingdom. Dr. Shen became a full professor of applied mechanics at Shanghai Jiao Tong University at the end of 1992. In 1995, he was invited again as a visiting professor at the University of Cardiff and in 1998–1999, as a visiting research fellow at the Hong Kong Polytechnic University, and in 2002–2003 as a visiting professor at the City University of Hong Kong. Also in 2002, he was a Tan Chin Tuan exchange fellow at the Nanyang Technological University in Singapore and in 2004, he was a Japan Society for the Promotion of Science (JSPS) invitation fellow at the Shizuoka University in Japan. In 2007, Dr. Shen was a visiting professor at the University of Western Sydney in Australia. Dr. Shen’s research interests include stability theory and, in general, nonlinear response of plate and shell structures. He has published over 190 journal papers, of which 123 are international journal papers. His research publications have been widely cited in the areas of computational mechanics and structural engineering (more than 1500 times by papers published in 387 international archival journals, and 220 local journals, excluding self-citations). He is the coauthor of the books Buckling of Structures (with T.-Y. Chen) and Postbuckling Behavior of Plates and Shells. He won the second Science and Technology Progress Awards of Shanghai in 1998 and 2003, respectively. Currently, Dr. Shen serves on the editorial boards of the journal Applied Mathematics and Mechanics (ISSN: 0253-4827) and the International Journal of Structural Stability and Dynamics (ISSN: 0219-4554). He is a member of the American Society of Civil Engineers.

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1 Modeling of Functionally Graded Materials and Structures

1.1

Introduction

The most lightweight composite materials with high strength=weight and stiffness=weight ratios have been used successfully in aircraft industry and other engineering applications. However, the traditional composite material is incapable to employ under the high-temperature environments. In general, the metals have been used in engineering ﬁeld for many years on account of their excellent strength and toughness. In the high-temperature condition, the strength of the metal is reduced similar to the traditional composite material. The ceramic materials have excellent characteristics in heat resistance. However, the applications of ceramic are usually limited due to their low toughness. Recently, a new class of composite materials known as functionally graded materials (FGMs) has drawn considerable attention. A typical FGM, with a high bending–stretching coupling effect, is an inhomogeneous composite made from different phases of material constituents (usually ceramic and metal). 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 isotropic matrix. Within FGMs the different microstructural phases have different functions, and the overall FGMs attain the multistructural status from their property gradation. By gradually varying the volume fraction of constituent materials, their material properties exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems and mitigating thermal stress concentrations. This is due to the fact that the ceramic constituents of FGMs are able to withstand high-temperature environments due to their better thermal resistance characteristics, while the metal constituents provide stronger mechanical performance and reduce the possibility 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 to develop high-resistant materials using FGMs had been continued. FGMs were initially designed as thermal barrier materials for aerospace structures and fusion reactors (Hirai and Chen 1999, Chan 2001, Uemura 2003). They 1

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Phase B particles with phase A matrix

Transition zone

FIGURE 1.1 An FGM with the volume fractions of constituent phases graded in one (vertical) direction. (From Yin, H.M., Sun, L.Z., and Paulino, G.H., Acta Mater., 52, 3535, 2004. With permission.)

Phase A particles with phase B matrix

are now developed for the general use as structural components in hightemperature environments. An example is FGM thin-walled rotating blades as shown in Figure 1.2 (Librescu and Song 2005). Potential applications of FGM are both diverse and numerous. Applications of FGMs have recently been reported in the open literature, e.g., FGM sensors (Müller 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

Y Ω Ro

g

y

X

xp

r1 d L

yD

g +b o x z, Z

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

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Modeling of Functionally Graded Materials and Structures

2 mm

3

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

aspects of FGMs have been published in the past few decades (Fuchiyama and Noda 1995, Markworth et al. 1995, Tanigawa 1995, Noda 1999, Paulino et al. 2003). They show that most of early research studies in FGMs had more focused on thermal stress analysis and fracture mechanics. A comprehensive survey for bending, buckling, and vibration analysis of plate and shell structures made of FGMs was presented by Shen (2004). Recently, Birman and Byrd (2007) presented a review of the principal developments in FGMs that 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 different properties. A detailed description of actual graded microstructures is usually not available, except perhaps for information on volume fraction distribution. Since the volume fraction of each phase gradually varies in the gradation direction, the effective properties of FGMs change along this direction. Therefore, we have two possible approaches to model FGMs. For the ﬁrst choice, a piecewise variation of the volume fraction of ceramic or metal is assumed, and the FGM is taken to be layered with the same volume fraction in each region, i.e., quasihomogeneous ceramic–metal layers (Figure 1.4a). For the second choice, a continuous variation of the volume fraction of ceramic or metal is assumed (Figure 1.4b), and the metal volume fraction can be represented as the following function of the thickness coordinate Z. 2Z þ h N Vm ¼ 2h

(1:1)

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

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(a)

(b)

FIGURE 1.4 Analytical model for an FGM layer.

FGM layer thickness. As is presented in Figure 1.5, changing the value of N generates an inﬁnite number of composition distributions. In order to accurately model the material properties of FGMs, the properties must be temperature- and position-dependent. This is achieved by using

1.00 N = 0.1 0.75

N = 0.2

Vm

N = 0.5 0.50

N = 1.0 N = 2.0 N = 3.0 N = 5.0

0.25

N = 10.0 N = 100 0.00 −0.50

−0.25

0.00 Z/h

FIGURE 1.5 Volume fraction of metal along the thickness.

0.25

0.50

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Modeling of Functionally Graded Materials and Structures

5

a simple rule of mixture of composite materials (Voigt model). The effective material properties Pf of the FGM layer, like Young’s modulus Ef, and thermal expansion coefﬁcient af, can then be expressed as X Pf ¼ Pj Vfj (1:2) j¼1

where Pj and Vfj are the material properties and volume fraction of the constituent material j, and the sum of the volume fractions of all the constituent materials makes 1, i.e., X Vfj ¼ 1 (1:3) j¼1

Since functionally graded structures are most commonly used in hightemperature environment where signiﬁcant changes in mechanical properties of the constituent materials are to be expected (Reddy and Chin 1998), it is essential to take into consideration this temperature-dependency for accurate prediction of the mechanical response. Thus, the effective Young’s modulus Ef, Poisson’s ratio nf, thermal expansion coefﬁcient af, and thermal conductivity kf are assumed to be temperature dependent and can be expressed as a nonlinear function of temperature (Touloukian 1967): Pj ¼ P0 (P1 T 1 þ 1 þ P1 T þ P2 T 2 þ P3 T 3 )

(1:4)

where P0, P1, P1, P2, and P3 are the coefﬁcients of temperature T (in K) and are unique to the constituent materials. Typical values for Young’s modulus Ef (in Pa), Poisson’s ratio nf, thermal expansion coefﬁcient af (in K1), and the thermal conductivity kf (in W mK1) of ceramics and metals are listed in Tables 1.1 through 1.4 (from Reddy and Chin 1998). From Equations 1.1 through 1.3, one has (Gibson et al. 1995): Ef (Z, T) ¼ [Em (T) Ec (T)]

2Z þ h N þ Ec (T) 2h

(1:5a)

TABLE 1.1 Modulus of Elasticity of Ceramics and Metals in Pa for Ef P0

P1

P1

Zirconia

244.27e þ 9

0

1.371e 3

1.214e 6

3.681e 10

Aluminum oxide

349.55e þ 9

0

3.853e 4

4.027e 7

1.673e 10

Silicon nitride

348.43e þ 9

0

3.070e 4

2.160e 7

8.946e 11

Ti-6Al-4V Stainless steel

122.56e þ 9 201.04e þ 9

0 0

4.586e 4 3.079e 4

0 6.534e 7

0 0

Nickel

223.95e þ 9

0

2.794e 4

3.998e 9

0

Materials

P2

P3

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

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TABLE 1.2 Coefﬁcient of Thermal Expansion of Ceramics and Metals in K1 for af Materials Zirconia

P0

P1

P1

12.766e 6

0

1.491e 3

P2

P3

1.006e 5

6.778e 11

Aluminum oxide

6.8269e 6

0

1.838e 4

0

0

Silicon nitride

5.8723e 6

0

9.095e 4

0

0

Ti-6Al-4V

7.5788e 6

0

6.638e 4

12.330e 6 9.9209e 6

0 0

8.086e 4 8.705e 4

Stainless steel Nickel

3.147e 6

0

0 0

0 0

Source: 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 Materials Zirconia Aluminum oxide Silicon nitride Ti-6Al-4V Stainless steel Nickela Nickelb

P0 1.7000 14.087 13.723 1.0000 15.379 187.66 58.754

P1

P1 1.276e 4

0 1123.6 0

P2

P3

6.648e 8

6.227e 3

0

1.032e 3

5.466e 7

0

1.704e 2

0

1.264e 3

0 0 7.876e 11

0

0

2.092e 6

7.223e 10

0

2.869e 3

4.005e 6

1.983e 9

0

4.614e 4

6.670e 7

1.523e 10

Source: Reddy, J.N. and Chin, C.D., J. Therm. Stresses, 21, 593, 1998. With permission. For 300 K T 635 K. b For 635 K T. a

TABLE 1.4 Poisson’s Ratio of Ceramics and Metals for nf P0

P1

Zirconia

0.2882

0

1.133e 4

0

0

Aluminum oxide

0.2600

0

0

0

0 0

Materials

P1

P2

P3

Silicon nitride

0.2400

0

0

0

Ti-6Al-4V

0.2884

0

1.121e 4

0

0

Stainless steel

0.3262

0

2.002e 4

3.797e 7

0

Nickel

0.3100

0

0

0

0

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

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Modeling of Functionally Graded Materials and Structures 2Z þ h N þ ac (T) 2h 2Z þ h N þ kc (T) kf (Z, T) ¼ [km (T) kc (T)] 2h 2Z þ h N þ nc (T) nf (Z, T) ¼ [nm (T) nc (T)] 2h

7

af (Z, T) ¼ [am (T) ac (T)]

(1:5b) (1:5c) (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 overall material properties and responses; however, owing to the assumed simpliﬁcations 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 the graded microstructures must be evaluated based on the volume fraction distribution and the approximate shape of the dispersed phase. Several micromechanics models have also been developed over the years to infer the effective properties of FGMs. The Mori–Tanaka scheme (Mori and Tanaka 1973, Benveniste 1987) for estimating the effective moduli is applicable to regions of the graded microstructure which have a well-deﬁned continuous matrix and a discontinuous particulate phase as depicted in Figure 1.1. It takes into account the interaction of the elastic ﬁelds among neighboring inclusions. It is assumed that the matrix phase, denoted by the subscript 1, is reinforced by spherical particles of a particulate phase, denoted by the subscript 2. In this notation, K1, G1, and V1 denote, respectively, the bulk modulus, the shear modulus, and the volume fraction of the matrix phase; K2, G2, and V2 denote the corresponding material properties and the volume fraction of the particulate phase. It should be noted that V1 þ V2 ¼ 1. The effective local bulk modulus Kf, the shear modulus Gf, thermal expansion coefﬁcient af, and thermal conductivity kf obtained by the Mori–Tanaka scheme for a random distribution of isotropic particles in an isotropic matrix are given by Kf K1 V2 ¼ K2 K1 1 þ (1 V2 )ð3(K2 K1 )=(3K1 þ 4G1 )Þ

(1:6a)

Gf G1 V2 ¼ G2 G1 1 þ (1 V2 )ð(G2 G1 )=(G1 þ f1 )Þ

(1:6b)

af a1 (1=Kf ) (1=K1 ) ¼ a2 a1 (1=K2 ) (1=K1 )

(1:6c)

kf k1 V2 ¼ k2 k1 1 þ (1 V2 )ð(k2 k1 )=3k1 Þ

(1:6d)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

where f1 ¼

G1 (9K1 þ 8G1 ) 6(K1 þ 2G1 )

(1:7)

The self-consistent method (Hill 1965) assumes that each reinforcement inclusion is embedded in a continuum material whose effective properties are those of the composite. This method does not distinguish between matrix and reinforcement phases and the same overall moduli are predicted in another composite in which the roles of the phases are interchanged. This makes it particularly suitable for determining the effective moduli in those regions which have an interconnected skeletal microstructure as depicted in Figure 1.6. The locally effective elastic moduli by the self-consistent method are given by d V1 V2 ¼ þ Kf Kf K2 Kf K1

(1:8a)

h V1 V2 ¼ þ Gf Gf G2 Gf G1

(1:8b)

where d ¼ 3 5h ¼

Kf Kf þ (4=3)Gf

(1:9)

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

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Modeling of Functionally Graded Materials and Structures

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From Equation 1.8a, one has Kf ¼

1 4 Gf ðV1 =(K1 þ (4=3)Gf )Þ þ ðV2 =(K2 þ (4=3)Gf )Þ 3

(1:10)

and Gf is obtained by solving the following quartic equation: ½V1 K1 =(K1 þ 4Gf =3) þ V2 K2 =(K2 þ 4Gf =3) þ 5½V1 G2 =(Gf G2 ) þ V2 G1 =(Gf G1 ) þ 2 ¼ 0

(1:11)

Then, the effective Young’s modulus Ef and Poisson’s ratio nf can be found from Ef ¼ 9KfGf=3Kf þ Gf and nf ¼ (3Kf 2Gf)=2(3Kf þ Gf), respectively. A comparison between the Mori–Tanaka and self-consistent models and the ﬁnite element simulation of FGM was presented in Reuter et al. (1997) and Reuter and Dvorak (1998). The Mori–Tanaka model was shown to yield accurate prediction of the properties with a well-deﬁned continuous matrix and discontinuous inclusions, while the self-consistent model was better in skeletal microstructures characterized by a wide transition zone between the regions with predominance of one of the constituent phases.

1.3

Reddy’s Higher Order Shear Deformation Plate Theory

Reddy (1984a,b) developed a simple higher order shear deformation plate theory (HSDPT), in which the transverse shear strains are assumed to be parabolically distributed across the plate thickness. The theory is simple in the sense that it contains the same dependent unknowns as in the ﬁrst-order shear deformation plate theory (FSDPT), and no shear correction factors are required. 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 its origin at the corner of the plate on the midplane. Let U, V, and W be the plate displacements parallel to a right-hand set of axes (X, Y, Z), where X is longitudinal and Z is perpendicular to the plate. Cx and Cy are the midplane rotations of the normal about the Y and X axes, respectively. The displacement components are assumed to be of the following form: U1 ¼ U(X, Y, t) þ ZCx (X, Y, t) þ Z2 jx (X, Y, t) þ Z3 zx (X, Y, t)

(1:12a)

U2 ¼ V(X, Y, t) þ ZCy (X, Y, t) þ Z jy (X, Y, t) þ Z zy (X, Y, t)

(1:12b)

U3 ¼ W(X, Y, t)

(1:12c)

2

3

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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 Z ¼ h=2), the transverse shear strains «4 and «5 should also vanish there. That is h «5 X, Y, , t ¼ 0, 2

«4

h X, Y, , t 2

¼0

(1:13)

which imply the following conditions jx ¼ 0

(1:14a)

jy ¼ 0 4 @W zx ¼ 2 þ Cx 3h @X 4 @W zy ¼ 2 þ Cy 3h @Y

(1:14b) (1:14c) (1:14d)

Putting the above conditions in Equation 1.12 leads to the following displacement ﬁeld "

# 4 2 2 @W U1 ¼ U þ 2 Cx x Cx þ 3 h @X " 2 # 4 2 @W Cy þ U2 ¼ V þ 2 Cy x 3 h @Y U3 ¼ W

(1:15a) (1:15b) (1:15c)

in which x is a tracer. If x ¼ 1, Equation 1.15 is for the case of the HSDPT, which contains the same dependent unknowns (U, V, W, Cx, and Cy) as in the FSDPT. If x ¼ 0, Equation 1.15 is reduced to the case of the FSDPT. The strains of the plate associated with the displacement ﬁeld given in Equation 1.15 are «1 ¼ «01 þ Z k01 þ Z2 k21 «2 ¼ «02 þ Z k02 þ Z2 k22 «3 ¼ 0 «4 ¼ «04 þ Z2 k24 «5 ¼ «05 þ Z2 k25 «6 ¼ «06 þ Z k06 þ Z2 k26

(1:16)

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where 2 @U 1 @W @Cx 4 @Cx @ 2 W 0 2 , k1 ¼ x 2 þ ¼ , k1 ¼ þ @X @X @X 2 @X 3h @X2 ! 2 @Cy @V 1 @W 4 @Cy @ 2 W 0 0 2 þ , k2 ¼ x 2 þ «2 ¼ , k2 ¼ @Y 2 @Y 3h @Y2 @Y @Y @W 4 @W , k24 ¼ x 2 Cy þ «04 ¼ Cy þ @Y h @Y @W 4 @W 0 2 «5 ¼ C x þ , k5 ¼ x 2 Cx þ @X h @X «01

(1:17)

@U @V @W @W þ þ @Y @X @X @Y @Cx @Cy þ k06 ¼ @Y @X ! 4 @Cx @Cy @2W 2 þ þ2 k6 ¼ x 2 @Y @X 3h @X@Y «06 ¼

The plane stress constitutive equations may then be written in the form: 2 3 2 32 3 Q11 Q12 0 «1 s1 4 s2 5 ¼ 4 Q21 Q22 (1:18a) 0 54 «2 5 s6 «6 0 0 Q66 0 Q44 «4 s4 ¼ (1:18b) 0 Q55 «5 s5 where Qij are the transformed reduced stiffnesses deﬁned by Q11 ¼ Q22 ¼

Ef (Z, T) , 1 n2f

Q16 ¼ Q26 ¼ 0,

nf Ef (Z, T) , 1 n2f Ef (Z, T) ¼ Q66 ¼ 2(1 þ nf )

Q12 ¼

Q44 ¼ Q55

(1:19)

As in the classical plate theory, the stress resultants and couples are deﬁned by h=2 ð

(N i , Mi , Pi ) ¼

si (1, Z, Z3 )dZ,

i ¼ 1, 2, 6

(1:20a)

h=2 h=2 ð

(Q2 , R2 ) ¼

s4 (1, Z2 )dZ h=2

(1:20b)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells ðh (Q1 , R1 ) ¼

s5 (1, Z2 )dZ

(1:20c)

h

where N i and Qi are the membrane and transverse shear forces Mi is the bending moment per unit length Pi and Ri are the higher order bending moment and shear force, respectively Substituting Equation 1.18 into Equation 1.20, and taking Equation 1.16 into account, yields the constitutive relations of the plate 2 3 2 32 0 3 A B E « N 4 M 5 ¼ 4 B D F 54 k0 5 (1:21a) E F H k2 P A D «0 Q (1:21b) ¼ D F k2 R where Aij, Bij, etc. are the plate stiffnesses, deﬁned by þh=2 ð

(Aij , Bij , Dij , Eij , Fij , Hij ) ¼

(Qij )(1, Z, Z2 , Z3 , Z4 , Z6 )dZ, i, j ¼ 1, 2, 6 (1:22a) h=2 þh=2 ð

(Aij , Dij , Fij ) ¼

(Qij )(1, Z2 , Z4 )dZ,

i, j ¼ 4, 5

(1:22b)

h=2

The Hamilton principle for an elastic body is ðt2 (dU þ dV dK)dt ¼ 0

(1:23)

t1

where dU is the virtual strain energy dV is the virtual work done by external forces dK is the virtual kinetic energy ð ð h=2 dU ¼

(si d«i )dZ dX dY V h=2

ð

¼

(N i d«0i þ Mi dk0i þ Pi dk2i )dZ dX dY, V

i ¼ 1, 2, 6

(1:24a)

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ð dV ¼ [q(X, Y)dU3 ]dX dY

(1:24b)

V

ð h=2 ð dK ¼

_ j dU _ j )dZ dX dY, r(U

j ¼ 1, 2, 3

(1:24c)

V h=2

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

@N 1 @N 6 @2U @ 2 Cx @3W c I þ ¼ I1 2 þ I 2 1 4 @X @Y @t2 @t @X@t2

@ 2 Cy @N 6 @N 2 @2V @3W c I þ ¼ I1 2 þ I 2 1 4 @X @Y @t2 @t @Y@t2 @Q1 @Q2 @ @W @W @ @W @W þ þ þ N6 þ þ N2 dW: N1 N6 @X @Y @X @X @Y @Y @X @Y 2 @R1 @R2 @ P1 @ 2 P6 @ 2 P2 þ c1 þ q c2 þ2 þ þ 2 @X @Y @X @X@Y @Y2 ! @2W @2 @2W @2W @ 2 @U @V @ 2 @Cx @Cy 2 þ þ þ c1 I 5 2 ¼ I1 2 c1 I7 2 þ c1 I4 2 þ @t @X2 @t @X @Y @t @X @Y @t @Y2 @M1 @M6 @P1 @P6 @2U @ 2 Cx @3W þ Q 1 þ C2 R1 C1 þ ¼ I2 2 þ I3 dCx : c1 I 5 2 @X @Y @X @Y @t @t @X@t2 @ 2 Cy @M6 @M2 @P6 @P2 @2V @3W þ Q2 þ c2 R2 c1 þ ¼ I2 2 þ I3 dCy : c1 I 5 2 @X @Y @X @Y @t @t @Y@t2 dV:

(1:25)

where c1 ¼ 4=3h2, c2 ¼ 3c1, and I 2 ¼ I2 c1 I4 ,

I 5 ¼ I5 c1 I7 ,

I 3 ¼ I3 2c1 I5 þ c21 I7 ,

I8 ¼ I3 þ I5

(1:26a)

and the inertias Ii (i ¼ 1, 2, 3, 4, 5, 7) are deﬁned by h=2 ð

(I1 , I2 , I3 , I4 , I5 , I7 ) ¼

r(Z)(1, Z, Z2 , Z3 , Z4 , Z6 )dZ

(1:26b)

h=2

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

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Generalized Kármán-Type Nonlinear Equations

Based on Reddy’s HSDPT with a von Kármán-type of kinematic nonlinearity (Reddy 1984b) and including thermal effects, Shen (1997) derived a set of general von Kármán-type equations which can be expressed in terms of a stress function F, two rotations Cx and Cy, and a transverse displacement W, along with the initial geometric imperfection W*. These equations are then extended to the case of shear deformable FGM plates. Let F (X, Y) be the stress function for the stress resultants deﬁned by N x ¼ F,yy, N y ¼ F,xx, and N xy ¼ F,xy, where a comma denotes partial differentiation with respect to the corresponding coordinates. If thermal effect is taken into account, we assume T

N* ¼ N N ,

T

T

M* ¼ M M , P* ¼ P P

(1:27)

where N T, MT, ST, and PT are the forces, moments, and higher order moments caused by elevated temperature, and are deﬁned by 2

T

Nx

6 T 6 6 Ny 4 T N xy

3

3 2 Ax þh=2 ð 7 T 7 6A 7 Py 7 ¼ 4 y 5(1, Z, Z3 )DT(X, Y, Z)dZ 5 T h=2 Axy Pxy 2 T3 2 T3 2 T3 Sx Mx Px 6 T7 6 T7 7 4 6 6 7 6 7 6 T7 6 Sy 7 ¼ 6 My 7 2 6 Py 7 4 5 4 5 3h 4 5 T T T Mxy Pxy Sxy

T

T

Mx

Px

T

My

T Mxy

(1:28a)

(1:28b)

where DT(X, Y, Z) ¼ T(X, Y, Z) T0 is temperature rise from the reference temperature T0 at which there are no thermal strains, and 2

3 2 Ax Q11 4 Ay 5 ¼ 4 Q12 Axy Q16

Q12 Q22 Q26

32 1 Q16 Q26 54 0 0 Q66

3 0 a11 5 1 a22 0

(1:29)

where a11 and a22 are the thermal expansion coefﬁcients measured in the longitudinal and transverse directions, respectively. The partial inverse of Equation 1.21a yields 3 2 A* «0 4 M* 5 ¼ 4 (B*)T (E*)T P* 2

32 3 B* E* N* T 54 0 5 D* (F*) k F* H* k2

(1:30)

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where the superscript ‘‘T’’ represents the matrix transpose and in which the reduced stiffness matrices [A*ij], [Bij*], [Dij*], [Eij*], [Fij*], and [Hij*] (i,j ¼ 1, 2, 6) are functions of temperature and position, determined through relationships (Shen 1997): A* ¼ A1 ,

B* ¼ A1 B,

D* ¼ D BA1 B,

F* ¼ F EA1 B,

E* ¼ A1 E,

H* ¼ H EA1 E

(1:31)

From Equation 1.30, the bending moments, higher order moments, and transverse shear forces can be written in the form: * F,yy B21 * F,xx þ D11 * Cx ,x þ D12 * Cy ,y Mx ¼ M1 ¼ B11 T

* (Cx ,x þ W,xx ) þ F21 * (Cy ,y þ W,yy )] þ Mx c1 [F11

(1:32a)

* F,yy B22 * F,xx þ D12 * Cx ,x þ D22 * C y ,y My ¼ M2 ¼ B12 T

* (Cx ,x þ W,xx ) þ F22 * (Cy ,y þ W,yy )] þ My c1 [F12

(1:32b)

* F,xy þ D66 * (Cx ,y þ Cy ,x ) Mxy ¼ M6 ¼ B66 T

* (Cx ,y þ Cy ,x þ 2W,xy ) þ Mxy c1 F66

(1:32c)

* F,yy E21 * F,xx þ F11 * Cx ,x þ F12 * Cy ,y Px ¼ P1 ¼ E11 T

* (Cx ,x þ W,xx ) þ H12 * (Cy ,y þ W,yy )] þ Px c1 [H11

(1:32d)

* F,yy E22 * F,xx þ F21 * Cx ,x þ D22 * C y ,y Py ¼ P2 ¼ E12 T

* (Cx ,x þ W,xx ) þ H22 * (Cy ,y þ W,yy )] þ Py c1 [H12

(1:32e)

Q1 ¼ (A55 c2 D55 )(Cx þ W,x )

(1:32f)

R1 ¼ (D55 c2 F55 )(Cx þ W,x )

(1:32g)

Q2 ¼ (A44 c2 D44 )(Cy þ W,y )

(1:32h)

R2 ¼ (D44 c2 F44 )(Cy þ W,y )

(1:32i)

Substituting Equation 1.32 into Equation 1.25, and considering the condition of compatibility, which is also expressed in terms of F, Cx, Cy, W, and W*, the equations of motion are obtained in the following ~11 (W) L ~12 (Cx ) L ~13 (Cy ) þ L ~14 (F) L ~15 (N T ) L ~16 (MT ) L € , þC € , )þq € I (C ~ ~17 (W) ¼ L(W þ W*, F) þ L 8 x x y y

(1:33)

~ ~ 21 (F) þ L ~22 (Cx ) þ L ~23 (Cy ) L ~24 (W) L ~25 (N T ) ¼ 1 L(W L þ 2W*, W) (1:34) 2

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells ~31 (W) þ L ~32 (Cx ) L ~33 (Cy ) þ L ~34 (F) L ~35 (N T ) L ~36 (ST ) L € € I C ¼ I 5 W, 3 x x

(1:35)

~42 (Cx ) þ L ~43 (Cy ) þ L ~44 (F) L ~45 (N T ) L ~46 (ST ) ~41 (W) L L € € I C ¼ I 5 W, 3 y y

(1:36)

~ij() and the nonlinear operator L() ~ are deﬁned by where all linear operators L @4 @4 @4 ~ * * þ F21 * þ 4F66 * Þ 2 2 þ F22 * þ ðF12 L11 () ¼ c1 F11 @X4 @X @Y @Y4 @3 @3 ~12 () ¼ ðD11 * c1 F11 * Þ 3 þ ½ðD12 * þ 2D66 * Þ c1 ðF12 * þ 2F66 * Þ L @X @X@Y2 3 @ @3 ~13 () ¼ ½ðD12 * þ 2D66 * Þ c1 ðF21 * þ 2F66 * Þ 2 * c1 F22 *Þ 3 þ ðD22 L @X @Y @Y 4 4 4 @ @ @ ~14 () ¼ B21 * * þ B22 * 2B66 *Þ * L þ ðB11 þ B12 @X4 @X2 @Y2 @Y4 2 @2 @2 T T T T T ~15 (N T ) ¼ @ * * * * * B L B N þ B N N B N þ B N þ 2 þ 11 21 66 12 22 x y xy x y @X2 @X@Y @Y2 T 2 2 2 @ @ T T T ~16 M ¼ @ L Mx þ 2 Mxy þ 2 My @X2 @X@Y @Y 2 I4 I2 @ @2 ~ I1 þ L17 () ¼ c1 I5 I1 @X2 @Y2 @4 @4 @4 ~21 () ¼ A22 * * * * ð þ A Þ L þ 2A þ A 12 66 11 @X4 @X2 @Y2 @Y4 3 @ @3 ~22 () ¼ ðB21 * c1 E21 *Þ * B66 * Þ c1 ðE11 * E66 * Þ ½ ð L þ B 11 @X3 @X@Y2 3 @ @3 ~23 () ¼ ½ðB22 * B66 * Þ c1 ðE22 * E66 * Þ 2 * c1 E12 *Þ 3 þ ðB12 L @X @Y @Y 4 4 4 @ @ @ ~24 () ¼ c1 E21 * * þ E22 * 2E66 * Þ 2 2 þ E12 * L þ ðE11 @X4 @X @Y @Y4 2 2 @ @2 T T T T T ~25 (N T ) ¼ @ * * * * * L A A N þ A N N A N þ A N þ 12 22 66 11 12 x y xy x y @X2 @X@Y @Y2 3 @ ~31 () ¼ A55 2c2 D55 þ c2 F55 @ þ c1 ðF11 * c1 H11 *Þ 3 L 2 @X @X @3 * þ 2F66 * Þ c1 ðH12 * þ 2H66 * Þ þ c1 ½ðF21 @X@Y2 @2 ~32 () ¼ A55 2c2 D55 þ c2 F55 D11 * 2c1 F11 * þ c21 H11 * L 2 @X2 2 @ * 2c1 F66 * þ c21 H66 * D66 @Y2

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17

@2 ~33 () ¼ ðD12 * þ D66 * Þ c1 ðF12 * þ F21 * þ 2F66 * Þ þ c21 ðH12 * þ H66 *Þ L @X@Y ~ ~ L34 () ¼ L22 () h i i @ h T T T ~35 (N T ) ¼ @ ðB11 * c1 E11 * ÞN x þ ðB21 * c1 E21 * ÞN y þ * c1 E66 * ÞN xy ðB66 L @X @Y T T @ @ T ~36 (S ) ¼ S þ S L @X x @Y xy ~41 () ¼ A44 2c2 D44 þ c2 F44 @ þ c1 ½ðF12 * þ 2F66 *Þ L 2 @Y @3 @3 * þ 2H66 * Þ 2 * c1 H22 *Þ 3 c1 ðH12 þ c1 ðF22 @X @Y @Y ~42 () ¼ L ~33 () L @2 2 * ~43 () ¼ A44 2c2 D44 þ c2 F44 D66 * * 2c þ c L F H 1 66 66 2 1 @X2 2 @ * 2c1 F22 * þ c21 H22 * D22 @Y2 ~44 () ¼ L ~23 () L h i i @ h T T T ~45 (N T ) ¼ @ ðB66 * c1 E66 * ÞN xy þ * c1 E12 * ÞN x þ ðB22 * c1 E22 * ÞN y L ðB12 @X @Y @ @ T T T ~46 (S ) ¼ (S ) þ (S ) L @X xy @Y y 2 2 2 @2 @2 @2 ~ ¼ @ @ 2 @ þ L() (1:37) @X2 @Y2 @X@Y @X@Y @Y2 @X2 It is worthy to note that the governing differential equations (Equations 1.33 through 1.37) for an FGM plate are identical in form to those of unsymmetric cross-ply laminated plates. These general von Kármán-type equations will be used in solving many nonlinear problems, e.g., nonlinear bending, postbuckling, and nonlinear vibration of shear deformable FGM plates.

References Benveniste Y. (1987), A new approach to the application of Mori–Tanaka’s theory of composite materials, Mechanics of Materials, 6, 147–157. Birman V. and Byrd L.W. (2007), Modeling and analysis of functionally graded materials and structures, Applied Mechanics Reviews, 60, 195–216. Chan S.H. (2001), Performance and emissions characteristics of a partially insulated gasoline engine, International Journal of Thermal Science, 40, 255–261. Fuchiyama T. and Noda N. (1995), Analysis of thermal stress in a plate of functionally gradient material, JSAE Review, 16, 263–268.

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Gibson L.J., Ashby M.F., Karam G.N., Wegst U., and Shercliff H.R. (1995), Mechanical properties of natural materials. II. Microstructures for mechanical efﬁciency, Proceedings of the Royal Society of London Series A, 450, 141–162. Hill R. (1965), A self-consistent mechanics of composite materials, Journal of the Mechanics and Physics of Solids, 13, 213–222. Hirai T. and Chen L. (1999), Recent and prospective development of functionally graded materials in Japan, Materials Science Forum, 308–311, 509–514. Koizumi M. (1993), The concept of FGM, Ceramic Transactions, Functionally Gradient Materials, 34, 3–10. Librescu L. and Song S.-Y. (2005), Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability, Journal of Thermal Stresses, 28, 649–712. 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, Materials Science Forum, 423–425, 791–796. Markworth A.J., Ramesh K.S., and Parks W.P. (1995), Review: modeling studies applied to functionally graded materials, Journal of Material Sciences, 30, 2183–2193. Mori T. and Tanaka K. (1973), Average stress in matrix and average elastic energy of materials with misﬁtting inclusions, Acta Metallurgica, 2, 1571–574. Müller E., Drašar C., Schilz J., and Kaysser W.A. (2003), Functionally graded materials for sensor and energy applications, Materials Science and Engineering, A362, 17–39. Noda N. (1999), Thermal stresses in functionally graded material, Journal of Thermal Stresses, 22, 477–512. Paszkiewicz B., Paszkiewicz R., Wosko M., Radziewicz D., Sciana B., Szyszka A., Macherzynski W., and Tlaczala M. (2008), Functionally graded semiconductor layers for devices application, Vacuum, 82, 389–394. Paulino G.H., Jin Z.H., and Dodds Jr. R.H. (2003), Failure of functionally graded Materials, in Comprehensive Structural Integrity, Vol. 2 (eds. B. Karihallo and W.G. Knauss), Elsevier Science, New York, pp. 607–644. Qiu J., Tani J., Ueno T., Morita T., Takahashi H., and Du H. (2003), Fabrication and high durability of functionally graded piezoelectric bending actuators, Smart Materials and Structures, 12, 115–121. Reddy J.N. (1984a), A simple high-order theory for laminated composite plates, Journal of Applied Mechanics ASME, 51, 745–752. Reddy J.N. (1984b), A reﬁned nonlinear theory of plates with transverse shear deformation, International Journal of Solids and Structure, 20, 881–896. Reddy J.N. and Chin C.D. (1998), Thermoelastical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses, 21, 593–626. Reuter T. and Dvorak G.J. (1998), Micromechanical models for graded composite Materials: II. Thermomechanical loading, Journal of Mechanics and Physics of Solids, 46, 1655–1673. Reuter T., Dvorak G.J., and Tvergaard V. (1997), Micromechanical models for graded composite materials, Journal of Mechanics and Physics of Solids, 45, 1281–1302. Shen H.-S. (1997), Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis, Applied Mathematics and Mechanics, 18, 1137–1152. Shen H.-S. (2004), Bending, buckling and vibration of functionally graded plates and shells (in Chinese), Advances in Mechanics, 34, 53–60.

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Tanigawa Y. (1995), Some basic thermoelastic problems for nonhomogeneous structural materials, Applied Mechanics Reviews, 48, 287–300. Touloukian Y.S. (1967), Thermophysical Properties of High Temperature Solid Materials, McMillan, New York. Uemura S. (2003), The activities of FGM on new applications, Materials Science Forum, 423–425, 1–10. Vel S.S. and Batra R.C. (2002), Exact solution for thermoelastic deformations of functionally graded thick rectangular plates, AIAA Journal, 40, 1421–1433. Watari F., Yokoyama A., Omori M., Hirai T., Kondo H., Uo M., and Kawasaki T. (2004), Biocompatibility of materials and development to functionally graded implant for bio-medical application, Composites Science and Technology, 64, 893–908. Yamanoushi M., Koizumi M., Hiraii T., and Shiota I. (eds.) (1990), Proceedings of the First International Symposium on Functionally Gradient Materials, Japan. Yin H.M., Sun L.Z., and Paulino G.H. (2004), Micromechanics-based elastic model for functionally graded materials with particle interactions, Acta Materialia, 52, 3535–3543.

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2 Nonlinear Bending of Shear Deformable FGM Plates

2.1

Introduction

The nonlinear bending response of FGM plates subjected to transverse mechanical loads and thermal loading was the subject of recent investigations. Previous works for the linear bending of FGM rectangular, circular, and annular 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 deﬂection of heated FGM thin plates with Young’s modulus varying symmetrically to the middle plane in thickness direction. Suresh and Mortensen (1997) presented the large deformation problem of graded multilayered composites under thermomechanical loads. When the thermomechanical load reaches a high level, nonlinear strain–displacement relations have to be employed. As a result, a set of nonlinear equations will appear no matter what kind of analysis method is used. Based on the FSDPT, Praveen and Reddy (1998) analyzed nonlinear static and dynamic response of functionally graded ceramic–metal plates subjected to transverse mechanical loads and a onedimensional (1D) steady heat conduction by using ﬁnite element method (FEM). This work was then extended to the case of FGM square plates and shallow shell panels by Woo and Meguid (2001) using Fourier series technique, and to the case of FGM circular plates by Ma and Wang (2003) and Gunes and Reddy (2008), and to the case of FGM rectangular plates by GhannadPour and Alinia (2006) and Ovesy and GhannadPour (2007) using Ritz method and ﬁnite strip method, respectively. However, in their studies the formulations were based on the classical plate=shell theory, i.e., the theory based on the Kirchhoff–Love hypothesis and therefore the transverse shear deformations were not accounted for, and the material properties were assumed to be independent of temperature. Reddy (2000) developed theoretical formulations for thick FGM plates according to the HSDPT. In his study, both Navier solutions for linear bending of simply supported rectangular FGM plates and ﬁnite element models for nonlinear static and dynamic response were presented. The paper of Cheng (2001) also contains the 21

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

solution for nonlinear bending of transversely isotropic symmetric shear deformable FGM plates. Moreover, Shen (2002) provided a nonlinear bending analysis of simply supported shear deformable FGM rectangular plates subjected to a transverse uniform or sinusoidal load and in thermal environments. In his study, the material properties were considered to be temperature dependent and the effect of temperature rise on the nonlinear bending response was reported. Subsequently, Yang and Shen (2003a,b) developed a semianalytical-numerical method to examine the large deﬂection of thin and shear deformable FGM rectangular plates subjected to combined mechanical and thermal loads and under various boundary conditions. This method was then extended to the case of FGM hybrid plates with surface-bonded piezoelectric layers by Yang et al. (2004). In these studies, the material properties were assumed to be temperature independent and temperature dependent, respectively. Recently, Na and Kim (2006) studied nonlinear bending of clamped FGM rectangular plates subjected to a transverse uniform pressure and thermal loads by using a 3D FEM. In their study, the thermal loads were assumed as uniform, linear, and sinusoidal temperature rises across the thickness direction, whereas the material properties were assumed to be temperature independent. On the other hand, ceramics and the metals used in FGMs do store different amounts of heat, and therefore the heat conduction usually occurs (Tanigawa et al. 1996, Kim and Noda 2002). This leads to a nonuniform distribution of temperature through the plate thickness, but it is not accounted for in the above studies. This is because when the material properties are assumed to be functions of temperature and position, and the temperature is also assumed to be a function of position, the problem becomes very complicated. More recently, Shen (2007) provided a nonlinear thermal bending analysis of simply supported shear deformable FGM rectangular plates due to heat conduction. In his study, both heat conduction and temperature-dependent material properties were taken into account.

2.2

Nonlinear Bending of FGM Plates under Mechanical Loads in Thermal Environments

Here, we consider an FGM plate of length a, width b, and thickness h, which is made from a mixture of ceramics and metals. We assume that the composition is varied from the top to the bottom surface, i.e., the top surface (Z ¼ h=2) of the plate is ceramic-rich whereas the bottom surface (Z ¼ h=2) is metal-rich. The plate is subjected to a transverse uniform load q ¼ q0 or a sinusoidal load q ¼ 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 Poisson’s ratio nf depends weakly on temperature change and is assumed to be a constant. We assume the volume fraction Vm follows a simple power law as expressed by Equation 1.1. According to mixture rules, the effective

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Nonlinear Bending of Shear Deformable FGM Plates

23

Young’s modulus Ef and thermal expansion coefﬁcient af of an FGM plate can be written as 2Z þ h N þ Ec (T) 2h 2Z þ h N þ ac (T) af (Z, T) ¼ [am (T) ac (T)] 2h Ef (Z, T) ¼ [Em (T) Ec (T)]

(2:1a) (2:1b)

It is evident that when Z ¼ h=2, Ef ¼ Ec and af ¼ ac, and when Z ¼ h=2, Ef ¼ Em and af ¼ am. In the case of a transverse static load applied at the top surface of an FGM plate, the general von Kármán-type equations (Equations 1.33 through 1.36) can be written in the simple form as ~12 (Cx ) L ~13 (Cy ) þ L ~14 (F) L ~15 (N T ) L ~16 (MT ) ¼ L(W,F) ~ ~11 (W) L þ q (2:2) L ~22 (Cx ) þ L ~23 (Cy ) L ~24 (W) L ~25 (N T ) ¼ 1 L(W,W) ~21 (F) þ L ~ L 2

(2:3)

~32 (Cx ) L ~33 (Cy ) þ L ~34 (F) L ~35 (N T ) L ~36 (ST ) ¼ 0 ~31 (W) þ L L

(2:4)

~41 (W) L ~42 (Cx ) þ L ~43 (Cy ) þ L ~44 (F) L ~45 (N T ) L ~46 (ST ) ¼ 0 L

(2:5)

Note that the geometric nonlinearity in the von Kármán sense is given in ~ in Equations 2.2 and 2.3, and the other linear operators L ~ ij() are terms of L() deﬁned by Equation 1.37, and the forces, moments, and higher order moments caused by elevated temperature are deﬁned 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 movable edges) and case 2 (referred to herein as immovable edges), will be considered. Case 1. The edges are simply supported and freely movable in both the X- 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 are X ¼ 0, a: W ¼ Cy ¼ 0

(2:6a)

ðb N x dY ¼ 0 (movable edges)

(2:6b)

U ¼ 0 (immovable edges)

(2:6c)

0

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Y ¼ 0, b: W ¼ Cx ¼ 0

(2:6d)

ða N y dX ¼ 0 (movable edges)

(2:6e)

V ¼ 0 (immovable edges)

(2:6f)

0

It is noted that the presence of stretching–bending coupling gives rise to bending curvatures under the action of in-plane loading, no matter how small these loads may be. In this situation, the boundary condition of zero bending moment cannot be incorporated accurately. Because immovable edges are considered in the present analysis, Mx ¼ 0 (at X ¼ 0, a) and My ¼ 0 (at Y ¼ 0, b) are not included in Equation 2.6, as previously shown in Kuppusamy and Reddy (1984) and Singh et al. (1994). The unit end-shortening relationships are Dx 1 ¼ a ab

ðb ða 0 0

@U dX dY @X

ðb ða ("

@2F @2F 4 @Cx * * * þ A12 þ B11 2 E11 @X @Y2 @X2 3h 0 0 # @Cy 4 4 @2W @2W * * * þ B12 2 E11 E12 þ E*12 @Y 3h2 3h @X2 @Y2 ) 2 1 @W T T * Ny A*11 N x þ A12 dX dY 2 @X

1 ¼ ab

Dy 1 ¼ b ab

A*11

ða ðb 0 0

1 ¼ ab

@V dYdX @Y

ða ðb (" 0 0

@2F @2F 4 @Cx * * * 2 E21 * A22 þ A12 þ B21 @X @X2 @Y2 3h

# @Cy 4 4 @2W @2W * 2 E22 * * 2 E21 þ B22 þ E*22 @Y 3h 3h @X2 @Y2 ) 2 1 @W T T A*12 N x þ A*22 N y dY dX 2 @Y

(2:7a)

(2:7b)

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Nonlinear Bending of Shear Deformable FGM Plates

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where Dx and Dy are plate end-shortening displacements in the X- and Y-directions. Having developed the theory, we are now in a position to solve Equations 2.2 through 2.5 with boundary condition (Equation 2.6). Before proceeding, it is convenient to ﬁrst deﬁne the following dimensionless quantities for such plates (with gijk in Equations 2.14 and 2.16 below are deﬁned as in Appendix A): x ¼ pX=a,

y ¼ pY=b,

b ¼ a=b,

* D*22 A*11 A*22 ]1=4 W ¼ W=[D11

F ¼ F=[D*11 D*22 ]1=2 , (Cx ,Cy ) ¼ (Cx ,Cy )a=p[D*11 D*22 A*11 A*22 ]1=4 g 14 ¼ [D*22 =D*11 ]1=2 , g 24 ¼ [A*11 =A*22 ]1=2 , g 5 ¼ A*12 =A*22 (g T1 ,g T2 ) ¼ (ATx ,ATy )a2 =p2 [D*11 D*22 ]1=2 , (gT3 ,g T4 ,g T6 ,g T7 ) ¼ (DTx ,DTy ,FTx ,FTy )a2 =p2 h2 D*11 , (Mx ,My ,Px ,Py ,MTx ,MTy ,PTx ,PTy ) T

T

T

T

* A*11A*22]1=4 ¼ (Mx ,My ,4Px =3h2 ,4Py =3h2 ,Mx ,My ,4Px =3h2 ,4Py =3h2 )a2 =p2 D*11 [D*11 D22

lq ¼ q0 a4 =p4 D*11 [D*11 D*22 A*11 A*22 ]1=4 , (dx ,dy ) ¼ (Dx =a,Dy =b)b2 =4p2 [D*11 D*22 A*11 A*22 ]1=2

(2:8)

where ATx (¼ATy ), DTx (¼DTy ), and FTx (¼FTy ) are deﬁned by "

ATx

DTx

FTx

ATy

DTy

FTy

#

h=2 ð

¼ h=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 ) þ g14 L14 (F) L16 (MT ) ¼ g14 b2 L(W,F) þ lq (2:10) 1 L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) g24 L24 (W) ¼ g24 b2 L(W,W) 2

(2:11)

L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) L36 (ST ) ¼ 0

(2:12)

L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) L46 (ST ) ¼ 0

(2:13)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

where 4 @4 @4 2 4 @ þ 2g b þ g b 112 114 @x4 @x2 @y2 @y4 3 3 @ @ L12 () ¼ g120 3 þ g 122 b2 @x @x@y2 @3 @3 L13 () ¼ g131 b 2 þ g133 b3 3 @x @y @y 4 @4 @ @4 L14 () ¼ g140 4 þ g 142 b2 2 2 þ g 144 b4 4 @x @x @y @y 2 2 2 @ @ @ (MT ) þ b2 2 (MTy ) L16 (MT ) ¼ 2 (MTx ) þ 2b @x @x@y xy @y @4 @4 @4 L21 () ¼ 4 þ 2g212 b2 2 2 þ g214 b4 4 @x @x @y @y 3 3 @ @ L22 () ¼ g220 3 þ g 222 b2 @x @x@y2 @3 @3 L23 () ¼ g231 b 2 þ g233 b3 3 @x @y @y 4 4 @ @ @4 L24 () ¼ g240 4 þ g 242 b2 2 2 þ g 244 b4 4 @x @x @y @y 3 3 @ @ @ þ g 310 3 þ g 312 b2 L31 () ¼ g31 @x @x@y2 @x @2 @2 L32 () ¼ g31 g 320 2 g 322 b2 2 @x @y 2 @ L33 () ¼ g331 b @x@y L34 () ¼ L22 () @ T @ (S ) þ b (STxy ) L36 (ST ) ¼ @x x @y @ @3 @3 L41 () ¼ g41 b þ g 411 b 2 þ g 413 b3 3 @x @y @y @y L42 () ¼ L33 ()

L11 () ¼ g110

L43 () ¼ g41 g 430

@2 @2 g 432 b2 2 2 @x @y

L44 () ¼ L23 () @ T @ (Sxy ) þ b (STy ) L46 (ST ) ¼ @x @y @2 @2 @2 @2 @2 @2 þ 2 2 L() ¼ 2 2 2 @x @y @x@y @x@y @y @x

(2:14)

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Nonlinear Bending of Shear Deformable FGM Plates

27

The boundary conditions of Equation 2.6 become x ¼ 0, p: W ¼ Cy ¼ 0 ðp b2 0

(2:15a)

@2F dy ¼ 0 (movable edges) @y2

(2:15b)

dx ¼ 0 (immovable edges)

(2:15c)

W ¼ Cx ¼ 0

(2:15d)

y ¼ 0, p: ðp 0

@2F dx ¼ 0 (movable edges) @x2

(2:15e)

dy ¼ 0 (immovable edges)

(2:15f)

and the unit end-shortening relationships become dx ¼

1 2 4p b2 g24

g 24

dy ¼

g 24

0 0

@2W @2W g611 2 þ g 244 b2 2 @x @y

1 4p2 b2 g24

ðp ðp @Cy @2F @2F @Cx g 224 b2 2 g 5 2 þ g 24 g511 þ g 233 b @x @y @y @x

) 1 @W 2 2 þ(g 24 g T1 g 5 g T2 )DT dxdy (2:16a) g 24 2 @x

ðp ðp 2 2 @Cy @ F @Cx 2@ F g b þ g g b þ g 5 24 220 522 @x2 @y2 @x @y 0 0

@2W @2W g240 2 þ g 622 b2 2 @x @y

) 1 @W 2 þ(g T2 g 5 g T1 )DT dy dx (2:16b) g 24 b2 2 @y

Applying Equations 2.10 through 2.16, the nonlinear bending response of a simply supported FGM plate subjected to a transverse uniform or sinusoidal load and in thermal environments is now determined by means of a two-step perturbation technique, for which the small perturbation parameter has no physical meaning at the ﬁrst step, and is then replaced by a dimensionless central deﬂection at the second step. The essence of this procedure, in the present case, is to assume that X X «j wj (x, y), F(x, y, «) ¼ «j fj (x, y) W(x, y, «) ¼ j¼1

Cx (x, y, «) ¼

X j¼1

j¼0 j

« cxj (x, y),

Cy (x, y, «) ¼

X j¼1

«j cyj (x, y),

lq ¼

X

«j lj

j¼1

(2:17)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

where « is a small perturbation parameter and the ﬁrst term of wj(x, y) is assumed to have the form: w1 (x, y) ¼ A(1) 11 sin mx sin ny

(2:18)

Then, we expand the thermal bending moments in the double Fourier sine series as "

MTx

STx

MTy

STy

#

" ¼ «

M(1) x

S(1) x

M(1) y

S(1) y

#

X i¼1,3,...

X 1 sin ix sin jy ij j¼1,3,...

(2:19)

(1) (1) (1) where M(1) x , My , Sx , and Sy and all coefﬁcients 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 which can be written, for example, as

O(«0): L14 ( f0 ) ¼ 0

(2:20a)

L21 ( f0 ) ¼ 0

(2:20b)

L34 ( f0 ) ¼ 0

(2:20c)

L44 ( f0 ) ¼ 0

(2:20d)

O(«1): L11 (w1 ) L12 (cx1 ) L13 (cy1 ) þ g 14 L14 ( f1 ) ¼ g 14 b2 L(w1 , f0 ) þ l1

(2:21a)

L21 ( f1 ) þ g 24 L22 (cx1 ) þ g24 L23 (cy1 ) g 24 L24 (w1 ) ¼ 0

(2:21b)

L31 (w1 ) þ L32 (cx1 ) L33 (cy1 ) þ g14 L34 ( f1 ) ¼ 0

(2:21c)

L41 (w1 ) L42 (cx1 ) þ L43 (cy1 ) þ g14 L44 ( f1 ) ¼ 0

(2:21d)

O(«2): L11 (w2 ) L12 (cx2 ) L13 (cy2 ) þ g14 L14 ( f2 ) ¼ g14 b2 [L(w2 , f0 ) þ L(w1 , f1 )] þ l2 (2:22a) 1 L21 ( f2 ) þ g24 L22 (cx2 ) þ g 24 L23 (cy2 ) g24 L24 (w2 ) ¼ g24 b2 L(w1 , w1 ) (2:22b) 2 L31 (w2 ) þ L32 (cx2 ) L33 (cy2 ) þ g14 L34 ( f2 ) ¼ 0

(2:22c)

L41 (w2 ) L42 (cx2 ) þ L43 (cy2 ) þ g14 L44 ( f2 ) ¼ 0

(2:22d)

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Nonlinear Bending of Shear Deformable FGM Plates

29

O(«3): L11 (w3 ) L12 (cx3 ) L13 (cy3 ) þ g14 L14 ( f3 ) ¼ g 14 b2 [L(w3 , f0 ) þ L(w2 , f1 ) þ L(w1 , f2 )] þ l3

(2:23a)

1 L21 ( f3 ) þ g 24 L22 (cx3 ) þ g24 L23 (cy3 ) g 24 L24 (w3 ) ¼ g 24 b2 L(w1 , w2 ) (2:23b) 2 L31 (w3 ) þ L32 (cx3 ) L33 (cy3 ) þ g 14 L34 ( f3 ) ¼ 0

(2:23c)

L41 (w3 ) L42 (cx3 ) þ L43 (cy3 ) þ g 14 L44 ( f3 ) ¼ 0

(2:23d)

To solve these perturbation equations of each order, the amplitudes of the terms 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 thirdorder asymptotic solutions can be obtained as h i h i (3) (3) 3 4 W ¼ « A(1) 11 sin mx sin ny þ « A13 sin mx sin 3ny þ A31 sin 3mx sin ny þ O(« ) h

i

h i (2) 2 Cx ¼ « C(1) 11 cos mx sin ny þ « C20 sin 2mx h i (3) þ «3 C(3) cos mx sin 3ny þ C cos 3mx sin ny þ O(«4 ) 13 31 h i h i (2) 2 Cy ¼ « D(1) 11 sin mx cos ny þ « D02 sin 2ny h i (3) þ «3 D(3) sin mx cos 3ny þ D sin 3mx cos ny þ O(«4 ) 13 31 h i 2 y2 (0) x (1) b þ « B sin mx sin ny F ¼ B(0) 00 00 11 2 2 2 2 (2) y (2) x (2) (2) 2 þ « B00 b00 þ B20 cos 2mx þ B02 cos 2ny 2 2 h i (3) þ «3 B(3) sin mx sin 3ny þ B sin 3mx sin ny þ O(«4 ) 13 31

(2:24)

(2:25)

(2:26)

(2:27)

Note that for boundary condition case 1, it is just necessary to take (i) B(i) 00 ¼ b00 ¼ 0 (i ¼ 0, 2) in Equation 2.27, so that the asymptotic solutions have a similar form, and lq ¼ «l1 þ «2 l2 þ «3 l3 þ O(«4 )

(2:28)

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

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

g05 (1) g24 n2 b2 (1) 2 g24 m2 (1) 2 A11 , B(2) A11 , B(2) A 20 ¼ 02 ¼ 2 g06 32m g 6 32n2 b2 g7 11 2 g04 g02 g05 mn2 b2 (2) A(1) C(1) g14 g 24 A(1) 11 ¼ m 11 , C20 ¼ g 14 g 24 g 220 11 2 g00 g00 g06 4ðg31 þ g320 4m Þg6 2 g03 g01 g05 m2 nb (2)

A(1) D(1) g14 g24 A(1) 11 ¼ nb 11 , D02 ¼ g 14 g 24 g 233 11 2 2 g00 g00 g06 4 g41 þ g432 4n b g 7 B(1) 11 ¼ g 24

(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 3 (1) (3) « þ W (x,y) A « þ W ¼ W (1) (x, y) A(1) 11 11

(2:30)

2 3 (2) A(1) A(1) þ l(3) A(1) þ lq ¼ l(1) q q 11 « þ lq 11 « 11 «

(2:31)

and

From Equations 2.30 and 2.31 the load–central deﬂection relationship can be written as 2 3 q 0 a4 (1) W (2) W (3) W þ A þ A þ þ A ¼ A(0) W W W W *h D11 h h h

(2:32)

Similarly, the bending moment–central deﬂection relationships can be written as 2 3 M x a2 W W W (1) (2) (3) þ A þ A þ ¼ A(0) þ A MX MX MX MX *h D11 h h h 2 3 M y a2 W W W (0) (1) (2) (3) ¼ AMY þ AMY þ AMY þ AMY þ * D11 h h h h

(2:33)

(2:34)

Equations 2.32 through 2.34 can be employed to obtain numerical results for the load–deﬂection and load–bending moment curves of an FGM plate subjected to a transverse uniform or sinusoidal load and in thermal environments. Zirconia and titanium alloys were selected for the two constituent materials 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 nonlinear function of temperature of Equation 1.4 (Touloukian 1967), and typical values for Young’s modulus Ef (in Pa) and thermal expansion coefﬁcient af

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Nonlinear Bending of Shear Deformable FGM Plates

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(in K1) of Zirconia and Ti-6Al-4V can be found in Tables 1.1 and 1.2 (from Reddy and Chin 1998). Poisson’s ratio nf is assumed to be a constant, and nf ¼ 0.28. The results presented herein are for ‘‘movable’’ in-plane boundary conditions, unless it is stated otherwise. The load–center deﬂection curves for a zirconia=aluminum square plate with different values of the volume fraction index subjected to a uniform transverse load are compared in Figure 2.1 with numerical results of Praveen and Reddy (1998), using their material properties, i.e., for aluminum E ¼ 70 GPa, n ¼ 0.3, a ¼ 23.0 106 8C1, and for zirconia E ¼ 151 GPa, n ¼ 0.3, and a ¼ 10.0 106 8C1. Note that in Figure 2.1 the volume fraction index N is deﬁned for Vc, and E0 is a referenced value of Young’s modulus, and E0 ¼ 70 GPa. Clearly, the comparison is reasonably well. Figure 2.2 gives the load–deﬂection and load–bending moment curves of ZrO2=Ti-6Al-4V square plate with different values of volume fraction index N ( ¼ 0, 0.5, 1.0, 2.0, 5.0, and 1) subjected to a uniform pressure and under thermal environmental condition DT ¼ 0 K. The results show that a fully titanium alloy plate (N ¼ 0) has highest deﬂection and lowest bending moment. It can also be seen that the plate has higher deﬂection and lower bending moment when it has lower volume fraction. This is expected because the metallic plate is the one with the lower stiffness than the ceramic plate. Figure 2.3 gives the load–deﬂection and load–bending moment curves of ZrO2=Ti-6Al-4V square plates subjected to a uniform pressure and under three sets of thermal environmental conditions, referred to as I, II, and III. For environmental condition I, DT ¼ 0 K; for environmental condition II, 0.6 Uniform load Zirconia/aluminum b =1.0, b/h = 20 0.4

1 2

W/h

1: Aluminum 2: N = 2.0

3

3: N = 1.0

4

4: N = 0.5

5

5: Zirconia

0.2

Present Praveen and Reddy (1998) 0.0

0

5

10 4

4

q0b /E0h

FIGURE 2.1 Comparisons of load–central deﬂection curves for a zirconia=aluminum square plate.

15

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32

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 2.0 Uniform load T0 = 300 K ZrO2/Ti-6Al-4V b = 1.0, b/h = 20

W/h

1.5

1.0

ZrO2 N = 5.0 N = 2.0 N = 1.0 N = 0.5 Ti-6Al-4V

0.5

0.0

50

0

100 4

(a)

150

4

q0b /E0h

3

2

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

2

Mxb /E0h

4

Uniform load T0 = 300 K

ZrO2 N = 5.0 N = 2.0 N = 1.0 N = 0.5 Ti-6Al-4V

1

0

0

(b)

50

q0b4/E0h4

100

150

FIGURE 2.2 Effect of volume fraction index N on the nonlinear bending behavior of ZrO2=Ti-6Al-4V square plates under uniform pressure: (a) load–central deﬂection; (b) load–bending moment.

DT ¼ 200 K; and for environmental condition III, DT ¼ 300 K. Because the thermal expansion at the top surface is higher than that at the bottom surface, this expansion results in an upward deﬂection. It is seen that the deﬂections are reduced, but the bending moments are increased with increases in temperature. Note that for environmental conditions II and III the deﬂections are close to each other when W=h > 1.5.

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Nonlinear Bending of Shear Deformable FGM Plates 2.0

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

W/h

1.5

1.0

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

I: ∆T = 0 K II: ∆T = 200 K III: ∆T = 300 K

0.5

0.0

33

0

50

100

150

4 4 q0b /E0h

(a)

3 Uniform load T0 = 300 K ZrO2/Ti-6Al-4V b = 1.0, b/h = 20

2

0

Mxb /E h

4

2

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

1 I: ∆T = 0 K II: ∆T = 200 K III: ∆T = 300 K

0 (b)

0

50

100 4

150

4

q0b /E0h

FIGURE 2.3 Effect of temperature rise on the nonlinear bending behavior of ZrO2=Ti-6Al-4V square plates under uniform pressure: (a) load–central deﬂection; (b) load–bending moment.

Figure 2.4 shows the effect of in-plane boundary conditions on the nonlinear bending behavior of ZrO2=Ti-6Al-4V square plates under two environmental conditions. To this end, the load–deﬂection and load–bending moment curves of ZrO2=Ti-6Al-4V square plates under ‘‘movable’’ and ‘‘immovable’’ in-plane boundary conditions are displayed. The results

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 3

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)

W/h

2

Uniform load T0 = 300 K

1: Movable edges 2: Immovable edges

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

1 I: ∆T = 0 K II: ∆T = 200 K 0

50

0

(a)

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)

4

3

2

150

4

q0b /E0h

4

Mxb /E0h

100 4

2

Uniform load T0 = 300 K

1: Movable edges 2: Immovable edges

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

I: ∆T = 0 K II: ∆T = 200 K

1

0 (b)

0

50

100

150

q b4/E0h4 0

FIGURE 2.4 Effect of in-plane boundary conditions on the nonlinear bending behavior of ZrO2=Ti-6Al-4V square plates under uniform pressure: (a) load–central deﬂection; (b) load–bending moment.

show that the plate with immovable edges will undergo less deﬂection with smaller bending moments. Figure 2.5 compares the load–deﬂection and load–bending moment curves of ZrO2=Ti-6Al-4V square plates under two cases of transverse loading conditions along with two environmental conditions. It can be seen that both

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

W/h

1.5

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)

1.0 1: Uniform load 2: Sinusoidal load

0.5

0.0

I: ∆T = 0 K II: ∆T = 200 K 50

0

(a)

q b4/E h4 0 0

100

150

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

Mx b2/E0h4

2.0

1.5

1.0 1: Uniform load 2: Sinusoidal load

0.5

0.0 (b)

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)

I: ∆T = 0 K II: ∆T = 200 K 0

50

4 4 q0b /E0h

100

150

FIGURE 2.5 Comparisons of nonlinear responses of ZrO2=Ti-6Al-4V square plates subjected to a uniform or sinusoidal load and in thermal environments: (a) load–central deﬂection; (b) load–bending moment.

load–deﬂection and load–bending moment curves of the plate subjected to a sinusoidal 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, Mxb2=E0h4, and q0b4=E0h4 denote the dimensionless central deﬂection of the plate, central bending moment, and lateral pressure, respectively, where E0 ¼ Young’s modulus of Ti-6Al-4V at T ¼ 300 K.

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2.3

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Nonlinear Thermal Bending of FGM Plates due to Heat Conduction

In the case of a temperature ﬁeld applied at the top and bottom surfaces of an FGM plate, thermal bending usually occurs due to heat conduction. In the present case, we assume that the effective Young’s modulus Ef, thermal expansion coefﬁcient af, and thermal conductivity kf are functions of temperature, so that Ef, af, and kf are both temperature- and position-dependent. The Poisson ratio nf depends weakly on temperature change and is assumed to be a constant. According to mixture rules, we have 2Z þ h N þ Ec (T) Ef (Z, T) ¼ [Em (T) Ec (T)] 2h 2Z þ h N þ ac (T) af (Z, T) ¼ [am (T) ac (T)] 2h 2Z þ h N þ kc (T) kf (Z, T) ¼ [km (T) kc (T)] 2h

(2:35a) (2:35b) (2:35c)

We assume that the temperature variation occurs in the thickness direction only and 1D temperature ﬁeld is assumed to be constant in the XY plane of the plate. In such a case, the temperature distribution along the thickness can be obtained by solving a steady-state heat transfer equation: d dT k ¼0 dZ dZ

(2:36)

Equation 2.36 is solved by imposing the boundary conditions T ¼ TU at Z ¼ h=2 and T ¼ TL at Z ¼ h=2. The solution of this equation, by means of polynomial series, is (Javaheri and Eslami 2002): T(Z) ¼ TU þ (TL TU )h(Z)

(2:37)

where TU and TL are the temperatures at top and bottom surfaces of the plate, and " 1 2Z þ h kmc 2Z þ h Nþ1 k2mc 2Z þ h 2Nþ1 þ h(Z) ¼ (N þ 1)kc (2N þ 1)k2c C 2h 2h 2h k3mc 2Z þ h 3Nþ1 k4mc 2Z þ h 4Nþ1 þ (3N þ 1)k3c (4N þ 1)k4c 2h 2h # 5Nþ1 k5mc 2Z þ h (2:38a) (5N þ 1)k5c 2h

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kmc k2mc k3mc k4mc k5mc þ þ (N þ 1)kc (2N þ 1)k2c (3N þ 1)k3c (4N þ 1)k4c (5N þ 1)k5c (2:38b)

where kmc ¼ km kc. In particular, for an isotropic material, Equation 2.37 may then be expressed as T(Z) ¼

TU þ TL TU TL þ Z 2 h

(2:39)

In the present case, since there is no transverse static load applied, the nonlinear governing equations can be written in dimensionless form as L11 (W) L12 (Cx ) L13 (Cy ) þ g14 L14 (F) ¼ g14 b2 L(W,F) þ L16 (MT )

(2:40)

1 L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) g24 L24 (W) ¼ g24 b2 L(W,W) 2

(2:41)

L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) ¼ L36 (ST )

(2:42)

L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) ¼ L46 (ST )

(2:43)

where all nondimensional linear operators Lij() and nonlinear operator L() are deﬁned as expressed by Equation 2.14. Note that Equation 2.9 is now redeﬁned by "

ATx

DTx

FTx

ATy

DTy

FTy

#

þh=2 ð "

DT ¼ h=2

Ax Ay

# (1, Z, Z3 )DT(X, Y, Z)dZ

(2:44)

where DT is a constant and is deﬁned by DT ¼ TU TL. When DT ¼ 0, ATx ¼ (ATy ), DTx ¼ (DTy ), and FTx ¼ (FTy ) can be found in Appendix B, and if DT 6¼ 0, then ATx , DTx , and FTx can be found in Appendix D. L16(Mp), L36(Sp), and L46(Sp) in Equations 2.40, 2.42, and 2.43 are now treated as ‘‘pseudoloads,’’ and we expand the thermal bending moments in the double Fourier sine series as "

MTx

STx

MTy

STy

#

" ¼

M0x

S0x

M0y

S0y

#

X i¼1,3,...

X 1 sin ix sin jy ij j¼1,3,...

where M0x , M0y , S0x , and S0y are given in detail in Appendix E.

(2:45)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

By using a two-step perturbation technique, we obtain up to third-order asymptotic solutions in the same form as expressed by Equations 2.24 through 2.27, from which we have 3 W (1) (3) (1) ¼ A(1) A « A A « þ 11 11 W W t

(2:46)

and the bending moments can be written as 2 3 M x a2 þ A(1) A(1) « þ A(2) A(1) « þA(3) A(1) « þ ¼ A(0) MX MX 11 MX 11 MX 11 * t D11

(2:47)

2 3 M y a2 (1) (1) (2) (1) (3) (1) ¼ A(0) þ A A « þ A A « þA A « þ 11 11 11 MY MY MY MY D*11 t

(2:48)

in Equations 2.46 through 2.48, A(2) « is taken as the second perturbation 11 parameter relating to the temperature variation, i.e., 2 3 A(2) 11 « ¼ l þ Q2 (l) þ Q3 (l) þ

(2:49)

All symbols used in Equations 2.46 through 2.49 are also described in detail in Appendix E. Equations 2.46 through 2.49 can be employed to obtain numerical results for the thermal load–deﬂection and thermal load–bending moment curves of an FGM plate under heat conduction. Silicon nitride and stainless steel were selected for the two constituent materials of the substrate FGM layer, referred to as Si3N4=SUS304, in the present examples. The material properties of these two constituents are assumed to be nonlinear function of temperature of Equation 1.4, and typical values for Young’s modulus Ef (in Pa), thermal expansion coefﬁcient af (in K1), and the thermal conductivity kf (in W mK1) of silicon nitride and stainless steel can be found in Tables 1.1 through 1.3. Poisson’s ratio nf is assumed to be a constant, and nf ¼ 0.28. For these examples, the lower surface is hold at a prescribed temperature of 300 K, so that DTL ¼ 0 and DT ¼ DTU. Figure 2.6 presents the thermal load–deﬂection and thermal load–bending moment curves for square FGM plates with different values of volume fraction index N ( ¼ 0, 0.2, 0.5, 1.0, 2.0, 5.0, and 1) under temperature variation DTU at upper surface. It can be seen that the deﬂection of FGM plates with lower values of volume fraction index N is positive (downward), whereas for the plate with higher values of N the deﬂection becomes negative. This is due to the fact that the thermal expansion coefﬁcient at the lower surface is larger than that experienced by the upper surface. The results show that the plate has higher bending moment (except for N ¼ 0 and N ¼ 0.2)

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Nonlinear Bending of Shear Deformable FGM Plates

39

1.0 Si3N4/SUS304

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

W (mm)

0.5

0.0 Si3N4 N = 5.0 N = 2.0 N = 1.0

−0.5

−1.0

N = 0.5 N = 0.2 SUS304 200

0

400

600

∆TU (K)

(a)

100 Si3N4/SUS304 b = 1.0, b/h = 20 T0 = TL = 300 K

Mx (N m/m)

50

0

Si3N4 N = 5.0 N = 2.0 N = 1.0

−50

−100 (b)

0

N = 0.5 N = 0.2 SUS304 200

∆TU (K)

400

600

FIGURE 2.6 Effect of volume fraction index N on the nonlinear bending behavior of Si3N4=SUS304 square plates due to heat conduction: (a) load–central deﬂection; (b) load–bending moment.

when it has lower volume fraction. It can also be seen that the bending moment for a fully stainless steel plate (N ¼ 0) changes from positive to negative when the temperature variation DTU > 150 K. The results reveal that the nonlinear bending responses of an FGM plate due to heat conduction are quite different to those of an FGM plate subjected to transverse mechanical loads.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 0.75 Si3N4/SUS304 b =1.0, b/h = 20 T0 = TL = 300 K

W (mm)

0.50

II & N = 2.0 II & N = 0.5

I: TD II: TID

I & N = 2.0 I & N = 0.5

0.25

0.00

− 0.25

200

0

400

600

∆TU (K)

(a)

100 Si3N4/SUS304 b =1.0, b/h = 20 T0 = TL = 300 K

Mx (N m)

75

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

I: TD II: TID

I & N = 0.5

50

25

0

−25 (b)

0

200

400

600

∆TU (K)

FIGURE 2.7 Effect of temperature-dependency on the nonlinear bending behavior of Si3N4=SUS304 square plates due to heat conduction: (a) load–central deﬂection; (b) load–bending moment.

Figure 2.7 presents the thermal load–deﬂection and thermal load–bending moment curves for square FGM plates with two values of volume fraction index N ¼ 0.5 and 2.0 under two cases of thermoelastic properties TD and TID. Here, TD and TID represent, respectively, the material properties are temperature dependent and temperature independent, i.e., in a ﬁxed temperature T0 ¼ 300 K. Great differences could be seen in these two cases and

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Nonlinear Bending of Shear Deformable FGM Plates

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0.50 Si3N4/SUS304 b =1.0, b/h = 20 T0 = TL = 300 K

W (mm)

0.25

I: Immovable edges II: Movable edges

0.00

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

−0.25

−0.50

200

0

400

600

∆TU (K)

(a)

100 Si3N4/SUS304 b = 1.0, b/h = 20 T0 = TL = 300 K

Mx (N m)

50

I: Immovable edges II: Movable edges

0

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

−50

−100 (b)

0

200

400

600

∆TU (K)

FIGURE 2.8 Effect of in-plane boundary conditions on the nonlinear bending behavior of Si3N4=SUS304 plates due to heat conduction: (a) load–central deﬂection; (b) load–bending moment.

we believe that the temperature dependency of FGMs could not be neglected in the thermal bending analysis. Figure 2.8 shows the effect of in-plane boundary conditions on the nonlinear thermal bending behavior of FGM plates with N ¼ 0.5 and 2.0. The thermal load–deﬂection and thermal load–bending moment curves of square FGM plates under ‘‘movable’’ and ‘‘immovable’’ in-plane boundary

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

conditions are displayed. The results show that the plate with immovable edges will undergo less deﬂection with larger bending moments. It can also be seen that for the plate with ‘‘movable’’ edges both deﬂection and bending moment are decreased by increasing the volume fraction index N.

References Cheng Z.-Q. (2001), Nonlinear bending of inhomogeneous plates, Engineering Structures, 23, 1359–1363. Cheng Z.-Q. and Batra R.C. (2000), Three-dimensional thermoelastic deformations of a functionally graded elliptic plate, Composites Part B, 31, 97–106. Chung Y.-L. and Chen W.-T. (2007), Bending behavior of FGM-coated and FGMundercoated plates with two simply supported opposite edges and two free edges, Composite Structures, 81, 157–167. Croce L.D. and Venini P. (2004), Finite elements for functionally graded Reissner-Mindlin plates, Computer Methods in Applied Mechanics and Engineering, 193, 705–725. GhannadPour S.A.M. and Alinia M.M. (2006), Large deﬂection behavior of functionally graded plates under pressure loads, Composite Structures, 75, 67–71. Gunes R. and Reddy J.N. (2008), Nonlinear analysis of functionally graded circular plates under different loads and boundary conditions, International Journal of Structural Stability and Dynamics, 8, 131–159. Javaheri R. and Eslami M.R. (2002), Thermal buckling of functionally graded plates, AIAA Journal, 40, 162–169. Kashtalyan M. (2004), Three dimensional elasticity solution for bending of functionally graded rectangular plates, European Journal of Mechanics A=Solids, 23, 853–864. Kim K.-S. and Noda N. (2002), A Green’s function approach to the deﬂection of a FGM plate under transient thermal loading, Archive of Applied Mechanics, 72, 127–137. Kuppusamy T. and Reddy J.N. (1984), A three-dimensional nonlinear analysis of cross-ply rectangular composite plates, Computers and Structures, 18, 263–272. Ma L.S. and Wang T.J. (2003), Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings, International Journal of Solids and Structures, 40, 3311–3330. Mizuguchi F. and Ohnabe H. (1996), Large deﬂections of heated functionally graded simply supported rectangular plates with varying rigidity in thickness direction, in Proceedings of the 11th Technical Conference of the American Society for Composites, October 7–9, Atlanta, Georgia, Technomic Publ. Co., Inc., Lancaster, PA, pp. 957–966. Na K.-S. and Kim J.-H. (2006), Nonlinear bending response of functionally graded plates under thermal loads, Journal of Thermal Stresses, 29, 245–261. Ovesy H.R. and GhannadPour S.A.M. (2007), Large deﬂection ﬁnite strip analysis of functionally graded plates under pressure loads, International Journal of Structural Stability and Dynamics, 7, 193–211. Praveen G.N. and Reddy J.N. (1998), Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates, International Journal of Solids and Structures, 35, 4457–4476.

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Reddy J.N. (2000), Analysis of functionally graded plates, International Journal for Numerical Methods in Engineering, 47, 663–684. Reddy J.N. and Cheng Z.-Q. (2001), Three-dimensional thermomechanical deformations of functionally graded rectangular plates, European Journal of Mechanics A=Solids, 20, 841–855. Reddy J.N. and Chin C.D. (1998), Thermoelastical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses, 21, 593–626. Reddy J.N., Wang C.M., and Kitipornchai S. (1999), Axisymmetric bending of functionally grade circular and annular plates, European Journal of Mechanics A=Solids, 18, 185–199. Shen H.-S. (2002), Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments, International Journal of Mechanical Sciences, 44, 561–584. Shen H.-S. (2007), Nonlinear thermal bending response of FGM plates due to heat conduction, Composites Part B, 38, 201–215. Singh G., Rao G.V., and Iyengar N.G.R. (1994), Geometrically nonlinear ﬂexural response characteristics of shear deformable unsymmetrically laminated plates, Computers and Structures, 53, 69–81. Suresh S. and Mortensen A. (1997), Functionally graded metals and metal-ceramic composites: Part 2. Thermomechanical behaviour, International Materials Reviews, 42, 85–116. Tanigawa Y., Akai T., Kawamura R., and Oka N. (1996), Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperaturedependent material properties, Journal of Thermal Stresses, 19, 77–102. Touloukian Y.S. (1967), Thermophysical Properties of High Temperature Solid Materials, Macmillan, New York. Vel S.S. and Batra R.C. (2002), Exact solution for thermoelastic deformations of functionally graded thick rectangular plates, AIAA Journal, 40, 1421–1433. Woo J. and Meguid S.A. (2001), Nonlinear analysis of functionally graded plates and shallow shells, International Journal of Solids and Structures, 38, 7409–7421. Yang J. and Shen H.-S. (2003a), Nonlinear analysis of functionally graded plates under transverse and in-plane loads, International Journal of Non-Linear Mechanics, 38, 467–482. Yang J. and Shen H.-S. (2003b), Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions, Composites Part B, 34, 103–115. Yang J., Kitipornchai S., and Liew K.M. (2004), Non-linear analysis of the thermoelectro-mechanical behaviour of shear deformable FGM plates with piezoelectric actuators, International Journal for Numerical Methods in Engineering, 59, 1605–1632.

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3 Postbuckling of Shear Deformable FGM Plates

3.1

Introduction

When a ﬂat plate is under the action of edge compression in its middle plane, the plate is deformed but remains completely ﬂat when the edge forces are sufﬁciently small unless there is an initial geometric imperfection. By increasing the load, a state is reached when the plate bends slightly. The in-plane compressive load which is just sufﬁcient to keep the plate in a slightly bent form is called the critical load or buckling load. Once the buckling load is exceeded, the load–deﬂection relationship exhibits a stable character due to membrane forces which come into play. Actually, the buckling mode will change in the postbuckling range. These changes occur when the energy stored in the plate is sufﬁcient to carry the plate from one buckled form to the other. To obtain an accurate analysis of FGM plates in a wide postbuckling range, the changes in buckling mode must be taken into account. In the usual postbuckling analysis, the buckling mode of the plate is assumed to remain unchanged. This is reasonable assumption in the immediate postbuckling range, e.g., the postbuckling load less than about three times the buckling load. Many studies have been reported on the buckling and postbuckling analysis of FGM plates subjected to mechanical or thermal loading. Among those, thermal and mechanical buckling of simply supported FGM rectangular plates was studied by Javaheri and Eslami (2002a–c) based on the classical and higher order shear deformation plate theories. Na and Kim (2004, 2006a,b) used solid ﬁnite elements to calculate buckling temperature of FGM plates with fully clamped edges. Najaﬁzadeh and Eslami (2002a,b) and Najaﬁzadeh and Heydari (2004a,b, 2008) considered axisymmetric buckling of simply supported and clamped circular FGM plates under a uniform temperature rise or a radial compression based on the ﬁrst order and higher order shear deformation theory, respectively. Ma and Wang (2003a,b) did the postbuckling analysis of simply supported and clamped FGM circular plates under a radial compression or nonlinear temperature change across the plate thickness based on the classical von Kármán plate theory. Subsequently, they gave the relationships between axisymmetric buckling solutions of FGM circular 45

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plates based on third-order plate theory and classical plate theory (Ma and Wang 2004). The effect of initial geometric imperfections on the postbuckling behavior of FGM circular plates subjected to mechanical edge loads and heat conduction was then studied by Li et al. (2007). Naei et al. (2007) calculated buckling loads of radially loaded FGM circular thin plates with variable thickness by using ﬁnite element method (FEM). Buckling of FGM plates without or with piezoelectric layers subjected to various nonuniform in-plane loads, along with heat and applied voltage, was considered by Chen and Liew (2004) and Chen et al. (2008) using the ﬁrst-order shear deformation theory. Ganapathi et al. (2006) and Ganapathi and Prakash (2006) presented the buckling loads for simply supported FGM skew plates subjected to in-plane mechanical loads and heat conduction. In their analysis, the material properties were based on the Mori–Tanaka scheme and rule of mixture, respectively. This work was then extended to the case of thermal postbuckling of FGM skew plates by Prakash et al. (2008). Furthermore, Yang and Shen (2003) studied the postbuckling behavior of FGM thin plates under fully clamped boundary conditions. This work was then extended to the case of shear deformable FGM plates with various boundary conditions and various possible initial geometric imperfections by Yang et al. (2006). Woo et al. (2005) studied the postbuckling behavior of FGM plates and shallow shells under edge compressive loads and a temperature ﬁeld based on the higher order shear deformation theory. Wu (2004) studied the thermal buckling behavior of simply supported FGM rectangular plates under uniform temperature rise and gradient through the thickness based on the ﬁrst-order shear deformation plate theory. Shariat and Eslami (2005, 2006) performed the thermal buckling of imperfect FGM rectangular plates under three types of thermal loading as uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise, based on the ﬁrst-order shear deformation plate theory and the classical thin plate theory, respectively. Wu et al. (2007) studied the postbuckling of FGM rectangular plates under various boundary conditions subjected to uniaxial compression or uniform temperature rise based on the ﬁrst-order shear deformation plate theory. In the above studies, however, the materials properties were virtually assumed to be temperature-independent (T-ID). Park and Kim (2006) presented thermal postbuckling and vibration of simply supported FGM plates with temperature-dependent (T-D) materials properties by using FEM. Shukla et al. (2007) studied the postbuckling of clamped FGM rectangular plates subjected to thermomechanical loads. In their analysis, the temperature-dependent materials properties were considered and the analytical approach was based on fast-converging Chebyshev polynomials. It has been pointed out by Shen (2002) that the governing differential equations for an FGM plate are identical in form to those of unsymmetric cross-ply laminated plates, and applying in-plane compressive edge loads to such plates will cause bending curvature to appear. Consequently, the bifurcation buckling did not exist due to the stretching=bending coupling effect, as previously

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Postbuckling of Shear Deformable FGM Plates

47

proved by Leissa (1986) and Qatu and Leissa (1993), and the solutions are physically incorrect for simply supported FGM rectangular plates subjected to in-plane compressive edge loads and=or temperature variation. Moreover, Liew et al. (2003, 2004) studied thermal postbuckling behavior of FGM hybrid plates with different kinds of boundary conditions. In their analysis, the material properties were assumed to be temperatureindependent and temperature-dependent, respectively, but their results were only for the simple thermal loading case of uniform temperature rise. They conﬁrmed that the FGM plates with all four edges simply supported (SSSS) have no bifurcation buckling temperature, even for the loading case of uniform temperature change. Obviously, when the FGM plate is geometrically midplane symmetric, as reported in Birman (1995) and Feldman and Aboudi (1997), a bifurcation buckling load under in-plane compressive edge loads and=or temperature variation does exist. Recently, Shen (2005) provided a postbuckling analysis for simply supported, midplane symmetric FGM plates with fully covered or embedded piezoelectric actuators subjected to the combined action of mechanical, thermal, and electronic loads. In his study, the material properties were considered to be temperature-dependent and the effect of temperature rise and applied voltage on the postbuckling response was reported. This work was then extended to the case of postbuckling analysis of sandwich plates with FGM face sheets subjected to mechanical and thermal loads (Shen and Li 2008). On the other hand, due to the temperature gradient the plate is subjected to additional moments along with the membrane forces and the problem cannot be posed as an eigenvalue problem, when the four edges of the plate are simply supported. Therefore, the bifurcation solutions for FGM plates subjected to transverse temperature variation, i.e., linear and=or nonlinear gradient through the thickness, may also be physically incorrect. More recently, Shen (2007b) provided a thermal postbuckling analysis for simply supported, midplane symmetric FGM plates under in-plane nonuniform parabolic temperature distribution and heat conduction, and concluded that for the case of heat conduction, the postbuckling path for geometrically perfect plates is no longer of the bifurcation type.

3.2

Postbuckling of FGM Plates with Piezoelectric Actuators under Thermoelectromechanical Loads

Here we consider two types of hybrid laminated plate, referred to as (P=FGM)S and (FGM=P)S, which consists of four plies and is midplane symmetric, as shown in Figure 3.1. The length, width, and total thickness of the hybrid laminated plate are a, b, and h. The thickness of the FGM layer is hf, whereas the thickness of the piezoelectric layer is hp. The substrate FGM

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells a t0 t1

b

h

hp

Piezoelectric layer

hf

FGM

X

t3 t4

FGM Y

(a)

t2

Piezoelectric layer

Z a t0

b hf hp

h

FGM Piezoelectric layer Piezoelectric piezoelectric layer layer FGM

Y

(b)

X

t1 t2 t3 t4

Z

FIGURE 3.1 Conﬁgurations of two types of hybrid laminated plates: (a) (P=FGM)S plate; (b) (FGM=P)S plate.

layer is made from a mixture of ceramics and metals, the mixing ratio of which is varied continuously and smoothly in the Z-direction. It is assumed that the effective Young’s modulus Ef and thermal expansion coefﬁcient af of the FGM layer are temperature-dependent, whereas Poisson’s ratio nf depends weakly on temperature change and is assumed to be a constant. We assume the volume fraction Vm follows a simple power law. According to rule of mixture, we have Z t1 N þ Ec (T) Ef (Z, T) ¼ [Em (T) Ec (T)] t2 t1 Z t1 N þ ac (T) af (Z, T) ¼ [am (T) ac (T)] t2 t1

(3:1a) (3:1b)

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Postbuckling of Shear Deformable FGM Plates

49

It is evident that when Z ¼ t1, Ef ¼ Ec and af ¼ ac, and when Z ¼ t2, Ef ¼ Em and af ¼ am. The plate is assumed to be geometrically imperfect, and is subjected to a compressive edge load in the X-direction combined with thermal and electric loads. Hence the general von Kármán-type equations (Equations 1.33 through 1.36) can be written in the simple form as ~12 (Cx ) L ~13 (Cy ) þ L ~14 (F) L ~15 (N p ) L ~16 (Mp ) ¼ L(W ~ þ W*, F) (3:2) ~11 (W) L L ~21 (F) þ L ~ ~22 (Cx ) þ L ~23 (Cy ) L ~24 (W) L ~25 (N p ) ¼ 1 L(W L þ 2W*, W) 2

(3:3)

~32 (Cx ) L ~33 (Cy ) þ L ~34 (F) L ~35 (N p ) L ~36 (Sp ) ¼ 0 ~31 (W) þ L L

(3:4)

~41 (W) L ~42 (Cx ) þ L ~43 (Cy ) þ L ~44 (F) L ~45 (N p ) L ~46 (Sp ) ¼ 0 L

(3:5)

~ are deﬁned by ~ij() and nonlinear operator L() where all linear operators L Equation 1.37. In the above equations, the equivalent thermopiezoelectric loads are deﬁned by 2

2 T3 2 E3 P 3 N N N 6 P7 6 T7 6 E7 6M 7 6M 7 6M 7 6 P 7¼6 T 7þ6 E 7 4P 5 4P 5 4P 5 p T E S S S

(3:6)

where N T, M T, S T, P T and N E, ME, SE, PE are the forces, moments, and higher order moments caused by the elevated temperature and electric ﬁeld, respectively. The temperature ﬁeld is assumed to be uniformly distributed over the plate surface and through the plate thickness. For the plate-type piezoelectric material, only the transverse direction electric ﬁeld component EZ is dominant, and EZ is deﬁned as EZ ¼ F,Z, where F is the potential ﬁeld. If the voltage applied to the actuator is in the thickness only, then EZ ¼

Vk hp

(3:7)

where Vk is the applied voltage across the kth ply. The forces and moments caused by elevated temperature or electric ﬁeld are deﬁned by 2

T

Nx

6 T 6 Ny 4 T N xy

T

Mx

T

My

T Mxy

3

2 3 Ax ðtk X T 7 6A 7 Py 7 4 y 5 (1, Z, Z3 )DT dZ 5¼ k¼1 T Axy k tk1 Pxy T

Px

(3:8a)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 2

3

2 T3 2 T3 M Px 6 T 7 6 xT 7 6 T7 4 6 7 6 6 7 6 Sy 7 ¼ 4 My 7 5 3h2 4 Py 5 4 5 T T T Mxy Pxy Sxy T

Sx

(3:8b)

and 2

E

Nx

6 E 6 Ny 4 E N xy

3

2 3 tk B ð x X Vk E E 7 6B 7 My Py 7 4 y 5 (1, Z, Z3 ) dZ 5¼ hp k¼1 E E Bxy k tk1 Mxy Pxy 2 3 2 2 E3 E E 3 Sx M Px 6 E 7 6 xE 7 6 E7 4 6 7 6 6 7 6 Sy 7 ¼ 4 My 7 5 3h2 4 Py 5 4 5 E E E Mxy Pxy Sxy E

Mx

E

Px

(3:8c)

(3:8d)

in which 2

3 2 Ax Q11 6A 7 6 4 y 5 ¼ 4 Q12 Axy Q16 2 3 2 Bx Q11 6B 7 6 4 y 5 ¼ 4 Q12 Bxy Q16

Q12

Q16

32

1

Q12 Q22

76 Q26 54 0 0 Q66 32 1 Q16 76 Q26 54 0

Q26

Q66

Q22 Q26

0

3 0 7 a11 15 a22 0 3 0 7 d31 15 d32 0

(3:9a)

(3:9b)

where a11 and a22 are the thermal expansion coefﬁcients measured in the longitudinal and transverse directions, respectively, d31 and d32 are the piezoelectric strain constants of a single ply, and Qij are the transformed elastic constants, deﬁned by 2

3 2 c4 Q11 6 7 6 2 2 6 Q12 7 6 c s 6 7 6 Q22 7 6 s4 6 7¼6 6 6 Q 7 6 c3 s 6 16 7 6 6 7 4 Q 5 4 cs3 26

Q66

c 2 s2

2c2 s2 s4 c 4 þ s4 c 2 s2 2 2 2c s c4 cs3 c3 s cs3 c3 s cs3 c3 s 2c2 s2 c 2 s2

3 4c2 s2 2 3 7 Q11 4c2 s2 7 76 Q 7 4c2 s2 76 12 7 76 7 2cs(c2 s2 ) 74 Q22 5 7 2cs(c2 s2 ) 5 Q66 (c2 s2 )2

(3:10a)

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Postbuckling of Shear Deformable FGM Plates 2

3 2 c2 Q44 6 7 6 4 Q45 5 ¼ 4 cs s2 Q55

3 s2 7 Q44 cs 5 Q55 c2

51

(3:10b)

where Q11 ¼

E11 E22 n21 E11 , Q22 ¼ , Q12 ¼ , (1 n12 n21 ) (1 n12 n21 ) (1 n12 n21 ) Q44 ¼ G23 , Q55 ¼ G13 , Q66 ¼ G12

(3:10c)

E11, E22, G12, G13, G23, n12 and n21 have their usual meanings, and c ¼ cos u, s ¼ sin u

(3:10d)

where u is the lamination angle with respect to the plate X-axis. Note that for an FGM layer, a11 ¼ a22 ¼ af is given in detail in Equation 3.1b, and Qij ¼ Qij, which is expressed in Equation 1.19. All four edges are assumed to be simply supported. Depending upon the in-plane behavior at the edges, two cases, case 1 (referred to herein as movable edges) and case 2 (referred to herein as immovable edges), will be considered. These correspond to the case when the motion of the unloaded edges in the plane tangent to the plate structure’s midsurface, normal to the respective edge is either unrestrained or completely restrained, respectively. As a result, we have Case 1: The edges are simply supported and freely movable in the in-plane directions. In addition the plate is subjected to uniaxial compressive edge loads. Case 2: All four edges are simply supported. Uniaxial edge loads are acting in the X-direction. The edges X ¼ 0, a are considered freely movable (in the in-plane direction), the remaining two edges being unloaded and immovable (i.e., prevented from moving in the Y-direction). For both cases, the associated boundary conditions are X ¼ 0, a: W ¼ Cy ¼ 0 N xy ¼ 0,

Mx ¼ Px ¼ 0

(3:11a) (3:11b)

ðb N x dY þ P ¼ 0 0

(3:11c)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Y ¼ 0, b: W ¼ Cx ¼ 0 N xy ¼ 0,

(3:11d)

My ¼ P y ¼ 0

(3:11e)

(movable edges)

(3:11f)

ða N y dX ¼ 0 0

V¼0

(immovable edges)

(3:11g)

where P is a compressive edge load in the X-direction, Mx and My are the bending moments, and Px and Py are the higher order moments, deﬁned by * F,yy B21 * F,xx þ B61 * F,xy þ D11 * Cx ,x þ D12 * Cy ,y þ D16 * (Cx ,y þ Cy ,x ) Mx ¼ B11 T

* (Cx ,x þ W,xx ) þ F21 * (Cy ,y þ W,yy ) þ F61 * (Cx ,y þ Cy ,x þ 2W,xy )] þ Mx c1 [F11

(3:12a) * F,yy B22 * F,xx þ B62 * F,xy þ D12 * Cx ,x þ D22 * Cy ,y þ D26 * (Cx ,y þ Cy ,x ) My ¼ B12 T

* (Cx ,x þ W,xx ) þ F22 * (Cy ,y þ W,yy ) þ F62 * (Cx ,y þ Cy ,x þ 2W,xy )] þ My c1 [F12

(3:12b) * F,yy E21 * F,xx þ E61 * F,xy þ F11 * Cx ,x þ F12 * Cy ,y þ F16 * (Cx ,y þ Cy ,x ) Px ¼ E11 T

* (Cx ,x þ W,xx ) þ H12 * (Cy ,y þ W,yy ) þ H16 * (Cx ,y þ Cy ,x þ 2W,xy )] þ Px c1 [H11

(3:12c) * F,yy E22 * F,xx þ E62 * F,xy þ F21 * Cx ,x þ F22 * Cy ,y þ F26 * (Cx ,y þ Cy ,x ) Py ¼ E12 T

* (Cx ,x þ W,xx ) þ H22 * (Cy ,y þ W,yy ) þ H26 * (Cx ,y þ Cy ,x þ 2W,xy )] þ Py c1 [H12

(3:12d)

The condition expressing the immovability condition V ¼ 0 (on Y ¼ 0, b) is fulﬁlled on the average sense as ða ðb 0 0

@V dY dX ¼ 0 @Y

(3:13)

This condition in conjunction with Equation 3.14b below provides the compressive stresses acting on the edges Y ¼ 0, b.

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53

The average end-shortening relationships are Dx 1 ¼ a ab

ðb ða 0 0

@U dX dY @X

ðb ða 1 @2F @2F @2F 4 @Cx * *12 *16 *11 *11 A11 þ A A E þ B ¼ @X ab @Y2 @X2 @X@Y 3h2 0 0 ! @Cy 4 4 @Cx @Cy * * * * þ B16 2 E16 þ þ B12 2 E12 @Y @Y @X 3h 3h 4 @2W @2W @2W * * * þ E þ 2E 2 E11 12 16 3h @X2 @Y2 @X@Y ) 2 1 @W @W @W* * p p p * N y þ A16 * Nxy dX dY (3:14a) A11 N x þ A12 2 @X @X @X Dy 1 ¼ b ab

ða ðb 0 0

@V dY dX @Y

ða ðb 1 @2F @2F @2F 4 @Cx * * * * * ¼ þ B A22 þ A A E 12 26 21 21 @X ab @X2 @Y2 @X@Y 3h2 0 0 ! @Cy 4 4 @Cx @Cy * * * * þ B26 2 E26 þ þ B22 2 E22 @Y @Y @X 3h 3h 2 2 4 @2W * @ W þ 2E * @ W * þ E 2 E21 26 22 3h @X2 @Y2 @X@Y ) 2 1 @W @W @W* p p p * N x þ A22 * N y þ A26 * Nxy A12 dY dX (3:14b) 2 @Y @Y @Y where Dx and Dy are plate end-shortening displacements in the X- and Y-directions. It is evident that the above equations involve the stretching=bending coupling, as predicted by Bij and Eij. As argued previously, even for an FGM plate with all four edges simply supported, no bifurcation buckling could occur. For this reason, we consider here geometrically midplane symmetric FGM plates with fully covered or embedded piezoelectric actuators. In such a case, the stretching=bending coupling is zero-valued, i.e., Bij ¼ Eij ¼ 0. As ~14 ¼ L ~15 ¼ L ~22 ¼ L ~23 ¼ L ~24 ¼ L ~34 ¼ L ~35 ¼ L ~44 ¼ L ~45 ¼ 0 in Equations 3.2 a result, L through 3.5.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells Introducing dimensionless quantities of Equation 2.8, and * D22 * ]1=2 , (g T1 , gT2 , gP1 , g P2 ) ¼ ATx , ATy , BPx , BPy a2 =p2 [D11 * D22 * ]1=2 lx ¼ Pb=4p2 [D11

(3:15)

in which "

ATx

#

"

BPx

DT dZ

(3:16a)

# tk " X ð Bx Vk dZ DV ¼ B y hp k¼1

(3:16b)

DT ¼

ATy

# tk " X ð Ax k¼1

#

BPy

tk1

Ay

tk1

k

k

The nonlinear equations (Equations 3.2 through 3.5) may then be written in dimensionless form as L11 (W) L12 (Cx ) L13 (Cy ) ¼ g 14 b2 L(W þ W*, F)

(3:17)

1 L21 (F) ¼ g24 b2 L(W þ 2W*, W) 2

(3:18)

L31 (W) þ L32 (Cx ) L33 (Cy ) ¼ 0

(3:19)

L41 (W) L42 (Cx ) þ L43 (Cy ) ¼ 0

(3:20)

where all nondimensional linear operators Lij() and nonlinear operator L() are deﬁned by Equation 2.14. The boundary conditions expressed by Equation 3.11 become x ¼ 0, p:

1 p

W ¼ Cy ¼ 0

(3:21a)

F,xy ¼ Mx ¼ Px ¼ 0

(3:21b)

ðp

@2F dy þ 4lx b2 ¼ 0 @y2

(3:21c)

W ¼ Cx ¼ 0

(3:21d)

F,xy ¼ My ¼ Py ¼ 0

(3:21e)

b2 0

y ¼ 0, p:

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Postbuckling of Shear Deformable FGM Plates ðp 0

55

@2F dx ¼ 0 (movable edges) @x2

(3:21f)

dy ¼ 0 (immovable edges)

(3:21g)

and the unit end-shortening relationships become ðp ðp (

@2F @2F 1 @W 2 @W @W* g 5 2 g 24 g 24 2 @y @x 2 @x @x @x 0 0 )

(3:22a) þ g224 gT1 g5 gT2 DT þ g 224 gP1 g5 gP2 DV dx dy

1 dx ¼ 2 2 4p b g24

g224 b2

ðp ðp (

2 2 @2F 1 @W @W* 2@ F 2 @W g b b g 24 b2 g 5 @x2 @y2 2 24 @y @y @y 0 0 þ(gT2 g5 gT1 )DT þ (g P2 g 5 g P1 )DV dy dx (3:22b)

1 dy ¼ 2 2 4p b g24

By virtue of the fact that DV and DT are assumed to be uniform, the thermopiezoelectric coupling in Equations 3.2 through 3.5 vanishes, but terms in DV and DT intervene in Equation 3.22. Applying Equations 3.17 through 3.22, the compressive postbuckling behavior of perfect and imperfect, FGM hybrid plates with piezoelectric actuators under thermoelectromechanical loads is now determined by means of a two-step perturbation technique. The essence of this procedure, in the present case, is to assume that W(x, y, «) ¼

X

«j wj (x, y),

F(x, y, «) ¼

X

j¼1

Cx (x, y, «) ¼

X

«j fj (x, y),

j¼0 j

« cxj (x, y),

Cy (x, y, «) ¼

j¼1

X

«j cyj (x, y)

(3:23)

j¼1

where « is a small perturbation parameter and the ﬁrst term of wj(x, y) is assumed to have the form w1 (x, y) ¼ A(1) 11 sin mx sin ny

(3:24)

and the initial geometric imperfection is assumed to have a similar form * sin mx sin ny ¼ «mA(1) W*(x, y, «) ¼ «a11 11 sin mx sin ny * =A(1) where m ¼ a11 11 is the imperfection parameter.

(3:25)

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Substituting Equation 3.23 into Equations 3.15 through 3.20, collecting the terms of the same order of «, we obtain a set of perturbation equations which can be written, for example, as O(«1 ): L11 (w1 ) L12 (cx1 ) L13 (cy1 ) ¼ g 14 b2 L(w1 þ W*, f0 )

(3:26a)

L21 (f1 ) ¼ 0

(3:26b)

L31 (w1 ) þ L32 (cx1 ) L33 (cy1 ) ¼ 0

(3:26c)

L41 (w1 ) L42 (cx1 ) þ L43 (cy1 ) ¼ 0

(3:26d)

O(«2 ): L11 (w2 ) L12 (cx2 ) L13 (cy2 ) ¼ g 14 b2 [L(w2 , f0 ) þ L(w1 þ W*, f1 )] (3:27a) 1 L21 ( f2 ) ¼ g24 b2 L(w1 þ 2W*, w1 ) 2

(3:27b)

L31 (w2 ) þ L32 (cx2 ) L33 (cy2 ) ¼ 0

(3:27c)

L41 (w2 ) L42 (cx2 ) þ L43 (cy2 ) ¼ 0

(3:27d)

By using Equations 3.24 and 3.25 to solve these perturbation equations of each order, the amplitudes of the terms wj(x, y), fj(x, y), cxj(x, y), and cyj(x, y) are determined step by step. As a result, up to fourth-order asymptotic solutions can be obtained. h i W ¼ « A(1) 11 sin mx sin ny h i (3) þ «3 A(3) sin mx sin 3ny þ A sin 3mx sin ny þ O(«5 ) (3:28) 13 31 2 2 2 y2 (0) x (2) y (2) x (2) (2) 2 F ¼ B(0) b þ « b þ B B cos 2mx þ B cos 2ny 00 00 00 00 20 02 2 2 2 2 y2 x2 (4) (4) b(4) þ B(4) þ «4 B(4) 00 00 20 cos 2mx þ B02 cos 2ny þ B22 cos 2mxcos 2ny 2 2 (4) (4) þ B(4) 40 cos 4mx þ B04 cos 4ny þ B24 cos 2mx cos 4ny 5 þ B(4) 42 cos 4mx cos 2ny þ O(« )

h i cos mx sin ny Cx ¼ « C(1) 11 h i (3) 3 5 þ « C(3) 13 cos mx sin 3ny þ C31 cos 3mx sin ny þ O(« ) h i Cy ¼ « D(1) sin mx cos ny 11 h i (3) 3 þ « D(3) sin mx cos 3ny þ D sin 3mx cos ny þ O(«5 ) 13 31

(3:29)

(3:30)

(3:31)

It is mentioned that all coefﬁcients in Equations 3.28 through 3.31 are related and can be expressed in terms of A(1) 11 , for example

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Postbuckling of Shear Deformable FGM Plates C(1) 11 ¼ m B(2) 20 ¼

g04 (1) A , g00 11

D(1) 11 ¼ nb

2 g 24 n2 b2 (1) (1 þ 2m) A , 11 32m2

g03 (1) A g00 11 B(2) 02 ¼

57

2 m2 (1) (1 þ 2m) A 11 32g24 n2 b2

(3:32)

Next, upon substitution of Equations 3.28 through 3.31 into the boundary conditions (Equations 3.21c and 3.22a), the postbuckling equilibrium path can be written as (2) 2 (4) 4 lx ¼ l(0) x þ lx Wm þ lx Wm þ

(3:33)

(2) 2 (4) 4 dx ¼ d(0) x þ dx W m þ dx W m þ

(3:34)

and

in which Wm is the dimensionless form of maximum deﬂection, which (i) is assumed to be at the point (x, y) ¼ (p=2m, p=2n) and l(i) x and dx (i ¼ 0, 2, 4, . . . ) are all temperature-dependent and given in detail in Appendix F. Equations 3.33 and 3.34 can be employed to obtain numerical results for the postbuckling load–deﬂection or load–end-shortening curves of simply supported shear deformable FGM plates with piezoelectric actuators subjected to uniaxial compression combined with thermal and electric loads. From Appendix F, the buckling load of a perfect plate can readily be obtained numerically, by setting m = 0 (or W*=h ¼ 0), while taking Wm ¼ 0 (or W=h ¼ 0). In such a case, the minimum buckling load is determined by applying Equation 3.33 for various values of the buckling mode (m, n), which determine the number of half-waves in the X- and Y-directions, respectively. For numerical illustrations, the thickness of the FGM layer hf ¼ 1 mm whereas the thickness of piezoelectric layers hp ¼ 0.1 mm, so that the total thickness of the plate h ¼ 2.2 mm. Two sets of material mixture for FGMs are considered. One is silicon nitride and stainless steel, referred to as Si3N4=SUS304, and the other is zirconium oxide and titanium alloy, referred to as ZrO2=Ti-6Al-4V. The material properties of these constituents are assumed to be nonlinear function of temperature of Equation 1.4, and typical values for Young’s modulus Ef (in Pa) and thermal expansion coefﬁcient af (in K1) of them can be found in Tables 1.1 and 1.2. Poisson’s ratio nf is assumed to be a constant, and nf ¼ 0.28. PZT-5A is selected for the piezoelectric layers. The material properties of which are assumed to be linear functions of temperature change, i.e., E11 (T) ¼ E110 (1 þ E111 DT), E22 (T) ¼ E220 (1 þ E221 DT) G12 (T) ¼ G120 (1 þ G121 DT), G13 (T) ¼ G130 (1 þ G131 DT), a11 (T) ¼ a110 (1 þ a111 DT), a22 (T) ¼ a220 (1 þ a221 DT)

G23 (T) ¼ G230 (1 þ G231 DT)

(3:35)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 2.5

Px/Pcr

2.0

Isotropic thin plate (n = 0.326) b =1.0, (m, n) = (1, 1)

1.5 1.0

Present, W ∗/h = 0 Present, W ∗/h = 0.1 Theoritical (Dym 1974) Experiments (Yamaki 1961)

0.5 0.0 0.0

0.5

1.0

1.5

2.0

W/h FIGURE 3.2 Comparisons of postbuckling load–deﬂection curves for isotropic thin plates under uniaxial compression.

where E110, E220, G120, G130, G230, a110, a220, E111, E221, G121, G131, G231, a111, a221 are constants. Typical values adopted, as given in Oh et al. (2000), E110 ¼ E220 ¼ 61 GPa, G120 ¼ G130 ¼ G230 ¼ 24.2 GPa, n12 ¼ 0.3, a110 ¼ a220 ¼ 0.9 106 K1 and d31 ¼ d32 ¼ 2.54 1010 m V1; and E111 ¼ 0.0005, E221 ¼ G121 ¼ G131 ¼ G231 ¼ 0.0002, a111 ¼ a221 ¼ 0.0005. The postbuckling load–deﬂection curves for perfect and imperfect, isotropic thin square plates (n ¼ 0.326) subjected to uniaxial compression are compared in Figure 3.2 with the analytical solutions of Dym (1974) and the experimental results of Yamaki (1961). These comparisons show that the results from the present method are in good agreement with the existing results. Tables 3.1 through 3.4 present the buckling loads Pcr (in kN) for perfect, moderately thick (b=h ¼ 20), (P=FGM)S and (FGM=P)S hybrid laminated plates with unloaded edges immovable and with different values of the volume fraction index N ( ¼ 0.0, 0.2, 0.5, 1.0, 2.0, and 5.0) subjected to uniaxial compression under three sets of temperature rise (DT ¼ 0, 100, 200 K). Here, TD represents material properties for both substrate FGM layer and piezoelectric layers are temperature-dependent. TD-F represents material properties of substrate FGM layer are temperature-dependent but material properties of piezoelectric layers are temperature-independent, i.e., E111 ¼ E221 ¼ G121 ¼ G131 ¼ G231 ¼ a111 ¼ a221 ¼ 0 in Equation 3.35. TID represents material properties for both piezoelectric layers and substrate FGM layer are temperature-independent, i.e., in a ﬁxed temperature T0 ¼ 300 K for FGM layer, as previously used in Yang and Shen (2003). The control voltages with the same sign are also applied to the upper, lower, or middle piezoelectric layers, and are referred to as VU, VL, and VM. Three electrical

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TABLE 3.1 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, (P=FGM)S Plates with a Substrate Made of Si3N4=SUS304 and with Unloaded Edges Immovable under Uniform Temperature Rise and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) DT (in K)

VU ¼ VL ¼ (in V)

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

131.6953 131.1302

146.1075 145.5423

157.9360 157.3708

166.5642 165.9990

171.3940 170.8288

(P=FGM)S, TID, (m, n) ¼ (1, 1) 0

500 0

115.2026 114.6376

þ500

114.0725

130.5651

144.9771

156.8056

165.4337

170.2635

100

500

93.3042

109.9468

124.9499

137.6636

147.3829

153.4537

0

92.7391

109.3817

124.3848

137.0984

146.8177

152.8885

þ500

92.1741

108.8166

123.8196

136.5332

146.2524

152.3232

500

71.4058

88.1983

103.7924

117.3912

128.2016

135.5134

0

70.8407

87.6332

103.2272

116.8260

127.6364

134.9482

þ500

70.2757

87.0681

102.6621

116.2608

127.0711

134.3829

(P=FGM)S, TD-F, (m, n) ¼ (1, 1) 100 500 90.7119

106.8642

121.4538

133.8422

143.3414

149.3148

0

90.1468

106.2991

120.8886

133.2770

142.7762

148.7495

þ500 500

89.5818 63.4408

105.7340 79.6302

120.3235 94.7616

132.7118 108.0310

142.2110 118.6513

148.1842 125.9229

200

200

0

62.8758

79.0652

94.1965

107.4658

118.0861

125.3576

þ500

62.3107

78.5001

93.6314

106.9006

117.5209

124.7924

(P=FGM)S, TD, (m, n) ¼ (1, 1) 100 500 90.4283

106.5790

121.1672

133.5544

143.0529

149.0257

0

89.8680

106.0187

120.6067

132.9939

142.4923

148.4651

þ500

89.3077

105.4583

120.0463

132.4334

141.9318

147.9045

500

62.8991

79.0828

94.2088

107.4735

118.0899

125.3583

0

62.3435

78.5270

93.6530

106.9176

117.5339

124.8023

þ500

61.7878

77.9712

93.0971

106.3616

116.9779

124.2462

200

loading cases are considered. Here VU ¼ VL ¼ 0 V (or VM ¼ 0 V) implies that the buckling occurs under a grounding condition. Two kinds of substrate FGM layers, i.e., Si3N4=SUS304 and ZrO2=Ti-6Al-4V are considered. It can be found that the buckling load of (P=FGM)S plate is lower than that of (FGM=P)S plate. It can be seen that, for the hybrid plates with Si3N4=SUS304 substrate, a fully metallic plate (N ¼ 0) has lowest buckling load and that the buckling load increases as the volume fraction index N increases. This is expected because the metallic plate has a lower stiffness than the ceramic plate. It is found that the increase is about þ65% for the (P=FGM)S plate, and about þ67% for the (FGM=P)S one, from N ¼ 0 to N ¼ 5, under temperature change DT ¼ 100 K. It can also be seen that the temperature reduces the

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 3.2 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, (FGM=P)S Plates with a Substrate Made of Si3N4=SUS304 and with Unloaded Edges Immovable under Uniform Temperature Rise and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) DT (in K)

VM (in V)

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0 212.0199

(FGM=P)S, TID, (m, n) ¼ (1, 1) 0

100

500

138.5191

159.3017

177.6971

193.1219

204.8333

0

137.9540

158.7366

177.1320

192.5568

204.2680

211.4547

þ500 500

137.3889 116.6206

158.1714 137.5532

176.5668 156.5396

191.9915 172.8495

203.7028 185.6520

210.8894 194.0796

0

116.0556

136.9881

155.9744

172.2843

185.0867

193.5144

þ500

115.4905

136.4229

155.4093

171.7191

184.5215

192.9491

500

94.7222

115.8047

135.3821

152.5771

166.4707

176.1393

0

94.1571

115.2396

134.8169

152.0119

165.9054

175.5741

þ500

93.5921

114.6745

134.2518

151.4467

165.3402

175.0088

(FGM=P)S, TD-F, (m, n) ¼ (1, 1) 100 500 113.5414

133.8505

152.2997

168.1724

180.6583

188.9140

0

112.9764

133.2854

151.7345

167.6072

180.0931

188.3488

þ500

112.4113

132.7203

151.1694

167.0420

179.5278

187.7835

500

85.3580

105.6978

124.6823

141.4299

155.0314

164.5812

0 þ500

84.7929 84.2279

105.1327 104.5676

124.1171 123.5520

140.8647 140.2995

154.4662 153.9010

164.0159 163.4507

(FGM=P)S, TD, (m, n) ¼ (1, 1) 100 500 113.5180 0 112.9577

133.8241 133.2637

152.2709 151.7105

168.1422 167.5817

180.6273 180.0668

188.8832 188.3226

þ500

112.3974

132.7033

151.1500

167.0212

179.5062

187.7620

500

85.3354

105.6667

124.6441

141.3861

154.9840

164.5320

0

84.7797

105.1109

124.0882

140.8302

154.4280

163.9759

þ500

84.2241

104.5552

123.5323

140.2742

153.8719

163.4198

200

200

200

buckling load when the temperature dependency is put into consideration. The percentage decrease is about 14% for the (P=FGM)S plate and about 12% for the (FGM=P)S one from temperature changes from DT ¼ 0 K to DT ¼ 100 K under the same volume fraction distribution N ¼ 2. Also the buckling loads under TD-F and TD cases are very close under the same volume fraction distribution and the same temperature change. In contrast, for the ZrO2=Ti-6Al-4V hybrid laminated plate, the buckling load is lower than that of the Si3N4=SUS304 hybrid laminated plate and erratic behavior can be observed in thermal loading conditions DT ¼ 100 and 200 K. It can also be seen that the control voltage has a very small effect on the buckling loads for hybrid laminated plates; this is because the piezoelectric layer is much thinner than the FGM substrate. Very high voltages will be able to inﬂuence

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TABLE 3.3 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, (P=FGM)S Plates with a Substrate Made of ZrO2=Ti-6Al-4V and with Unloaded Edges Immovable under Uniform Temperature Rise and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) DT (in K)

VU ¼ VL ¼ (in V)

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

500 0

64.0000 63.4356

72.9739 72.4094

80.8184 80.2538

87.2604 86.6957

91.9643 91.3996

94.6043 94.0395

þ500

62.8712

71.8449

79.6892

86.1310

90.8348

93.4747

500

57.5279

64.4956

70.0364

73.9762

76.0389

75.9364

0

56.9635

63.9311

69.4718

73.4115

75.4741

75.3716

þ500

56.3991

63.3666

68.9072

72.8468

74.9094

74.8068

500

51.0557

56.0172

59.2544

60.6920

60.1135

57.2686

0

50.4913

55.4527

58.6898

60.1273

59.5487

56.7037

þ500

49.9269

54.8882

58.1252

59.5626

58.9839

56.1389

(P=FGM)S, TD-F, (m, n) ¼ (1, 1) 100 500 54.8204

(P=FGM)S, TID, (m, n) ¼ (1, 1) 0

100

200

200

59.5851

62.9913

64.9500

65.2654

63.7199

0

54.2561

59.0207

62.4268

64.3854

64.7007

63.1551

þ500 500

53.6917 45.9786

58.4563 45.1848

61.8623 42.4283

63.8208 37.8905

64.1360 31.5562

62.5904 23.3350

0

45.4144

44.6205

41.8639

37.3260

30.9916

22.7704

þ500

44.8502

44.0561

41.2995

36.7615

30.4270

22.2057

(P=FGM)S, TD, (m, n) ¼ (1, 1) 100 500 54.5302

200

59.2967

62.7049

64.6657

64.9834

63.4404

0

53.9709

58.7372

62.1453

64.1060

64.4237

62.8805

þ500

53.4117

58.1778

61.5857

63.5464

63.8639

62.3206

500

45.4104

44.6291

41.8861

37.3619

31.0410

22.8328

0

44.8561

44.0746

41.3314

36.8071

30.4860

22.2777

þ500

44.3018

43.5201

40.7768

36.2523

29.9311

21.7227

the buckling response of the hybrid laminated plate. However, such high voltages cannot be applied, because they lead to a breakdown in the material properties. Then Tables 3.5 and 3.6 present the buckling loads Pcr for the same two types of hybrid plates with unloaded edges movable subjected to uniaxial compression under three sets of temperature rise (DT ¼ 0, 100, 200 K). The results show that the buckling load is decreased with increase in temperature, but is increased as volume fraction index N increases at the same temperature. The numerical results also conﬁrm that the control voltage has no effect on the buckling loads of hybrid laminated plates when the unloaded edges are movable.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 3.4 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, (FGM=P)S Plates with a Substrate Made of ZrO2=Ti-6Al-4V and with Unloaded Edges Immovable under Uniform Temperature Rise and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) DT (in K)

VM (in V)

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0 110.8339

(FGM=P)S, TID, (m, n) ¼ (1, 1) 0

100

500

70.7813

82.1124

92.1400

100.5459

106.9247

0

70.2169

81.5479

91.5754

99.9812

106.3599

110.2691

þ500 500

69.6525 64.3091

80.9834 73.6340

91.0108 81.3580

99.4165 87.2617

105.7951 90.9992

109.7042 92.1660

0

63.7448

73.0695

80.7934

86.6970

90.4345

91.6012

þ500

63.1804

72.5050

80.2288

86.1323

89.8697

91.0364

500

57.8370

65.1557

70.5760

73.9775

75.0738

73.4981

0

57.2726

64.5912

70.0113

73.4128

74.5090

72.9333

þ500

56.7082

64.0267

69.4467

72.8481

73.9443

72.3685

(FGM=P)S, TD-F, (m, n) ¼ (1, 1) 100 500 60.6917

77.4619

200

67.4217

72.6482

76.2438

77.9534

0

60.1274

66.8572

72.0837

75.6792

77.3888

76.8972

þ500

59.5631

66.2928

71.5192

75.1146

76.8241

76.3325

500

50.9401

51.8613

50.6936

47.5837

42.4644

35.1598

0 þ500

50.3759 49.8116

51.2970 50.7327

50.1292 49.5647

47.0192 46.4547

41.8998 41.3352

34.5951 34.0305

(FGM=P)S, TD, (m, n) ¼ (1, 1) 100 500 60.6634 0 60.1041

67.3937 66.8343

72.6214 72.0618

76.2187 75.6590

77.9306 77.3708

77.4421 76.8822

200

200

þ500

59.5448

66.2749

71.5022

75.0993

76.8111

76.3223

500

50.8947

51.8260

50.6700

47.5728

42.4668

35.1762

0

50.3404

51.2715

50.1153

47.0180

41.9118

34.6211

þ500

49.7861

50.7170

49.5606

46.4632

41.3569

34.0661

Figures 3.3 and 3.4 give, respectively, the postbuckling load–deﬂection and load–shortening curves for (P=FGM)S and (FGM=P)S hybrid laminated plates (b=h ¼ 40, N ¼ 0.2) with unloaded edges immovable subjected to uniaxial compression and three sets of electrical loading, VU ¼ VL (or VM) ¼ 500, 0, þ500 V, and under DT ¼ 0 and 100 K. It is evident that the buckling loads reduce as the temperature increases, and the postbuckling path becomes lower. It can be found that the control voltage has a small effect on the postbuckling behavior of the plate. It can be seen that minus control voltages increase the buckling load and decrease the postbuckled deﬂection at the same temperature rise, whereas the plus control voltages decrease the buckling load and induce more large postbuckled deﬂections.

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TABLE 3.5 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, Si3N4=SUS304 Plates with Piezoelectric Actuators and with Unloaded Edges Movable Subjected to Temperature Rise (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) DT (in K)

N ¼ 0.0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

100

146.8044 144.8755

167.9184 165.3297

186.3678 183.2028

201.5086 197.8707

212.5516 208.5686

218.7312 214.5550

200

141.2601

161.6779

179.5191

194.1613

204.8411

210.8181

100

144.5326

164.9874

182.8609

197.5289

208.2266

214.2126

200

140.5758

160.9947

178.8367

193.4790

204.1585

210.1347

(P=FGM)S, (m, n) ¼ (1, 1) TID TD-FGM TD

(FGM=P)S, (m, n) ¼ (1, 1) TID 176.6633 TD-FGM TD

203.2696

226.8185

246.5632

261.5528

270.7491

100

174.1111

199.8871

222.7014

241.8298

256.3509

265.2588

200

169.3277

195.0592

217.8333

236.9282

251.4242

260.3173

100

174.1070

199.8824

222.6962

341.8245

256.3456

265.2538

200

169.3195

195.0497

217.8229

236.9173

251.4135

260.3075

Figures 3.5 and 3.6 show the effect of the volume fraction index N ( ¼ 0.2, 1.0, and 5.0) on the postbuckling behavior of (P=FGM)S and (FGM=P)S hybrid laminated plates (b=h ¼ 40) with unloaded edges immovable subjected to uniaxial compression and three sets of electrical loading, and under

TABLE 3.6 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, ZrO2=Ti-6Al-4V Plates with Piezoelectric Actuators and with Unloaded Edges Movable Subjected to Temperature Rise Temperature Rise (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) DT (in K)

N ¼ 0.0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

(P=FGM)S, (m, n) ¼ (1, 1) TID TD-FGM TD

81.2698

92.7595

102.8020

111.0477

117.0674

120.4437

100

77.6612

87.2275

95.5896

102.4555

107.4673

110.2784

200

74.0524

82.3901

89.6789

95.6635

100.0321

102.4828

100 200

77.3189 73.3692

86.8857 81.7080

95.2482 88.9973

102.1141 94.9821

107.1257 99.3503

109.9364 101.8001

(FGM=P)S, (m, n) ¼ (1, 1) TID

89.9576

104.4662

117.3045

128.0650

136.2290

141.2302

TD-FGM

100 200

85.1835 80.4093

97.2669 90.9440

107.9604 100.2675

116.9225 108.0808

123.7195 114.0052

127.8799 117.6293

TD

100

85.1797

97.2626

107.9557

116.9175

123.7146

127.8754

200

80.4018

90.9355

100.2582

108.0712

113.9956

117.6204

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 150

(P/FGM)S T0 = 300 K, N = 0.2 b = 1.0, b/h = 40 (m, n) = (1, 1)

100 Px (kN)

I: ∆T = 0 K II: ∆T = 100 K

I

50 2

3

2

3

1

1: VU = VL = −500 V 2: VU = VL= 0 V 3: VU = VL = +500 V

II

1

Immovable edges W ∗/h = 0.0 W ∗/h = 0.1

(a)

0 0.0

150

0.5

1.0

(P/FGM)S T0 = 300 K, N = 0.2 b = 1.0, b/h = 40 (m, n) = (1, 1)

100

W (mm)

2.0

2.5

I II 1: VU = VL = −500 V

Px (kN)

I: ∆T = 0 K II: ∆T = 100 K

2: VU = VL = 0 V 3: VU = VL = +500 V Immovable edges

50 2 3 0 −0.2 (b)

1.5

W ∗/h = 0.0

1

W ∗/h = 0.1

0.0

0.2 ∆x (mm)

0.4

0.6

FIGURE 3.3 Thermopiezoelectric effects on the postbuckling behavior of (P=FGM)S plates with unload edges immovable: (a) load deﬂection; (b) load shortening.

DT ¼ 100 K. It can be seen that the increase of the volume fraction index N yields an increase of the buckling load and postbuckling strength. Figures 3.7 and 3.8 show the effect of temperature rise DT ( ¼ 0, 100, and 200 K) on the postbuckling behavior of (P=FGM)S and (FGM=P)S hybrid laminated plates (b=h ¼ 40) with unloaded edges movable and with N ¼ 0.2 and 2.0 subjected to uniaxial compression. It can be seen that both buckling load and postbuckling strength are decreased with increase in temperature. It can also be seen that the buckling load of hybrid laminated plates with immovable unloaded edges is lower than that of the plate with movable

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C003 Final Proof page 65

8.12.2008 11:46am Compositor Name: DeShanthi

Postbuckling of Shear Deformable FGM Plates 150

(FGM/P)S T0 = 300 K, N = 0.2 b = 1.0, b/h = 40 (m, n) = (1, 1) 1 3 2

I

I: ∆T = 0 K II: ∆T = 100 K

II

1: VM = −500 V

Px (kN)

100

65

2: VM = 0 V 50

3

2

3: VM = +500 V

1 Immovable edges

W ∗/h = 0.0 W ∗/h = 0.1 0 0.0

0.5

1.0

(a) 150

(FGM/P)S T0 = 300 K, N = 0.2 b = 1.0, b/h = 40 (m, n) = (1, 1)

100

1.5 W (mm)

II

2.5

I

Px (kN)

I: ∆T = 0 K II: ∆T = 100 K

1: VM = −500 V 2: VM = 0 V 3: VM = +500 V Immovable edges

50

3 0 −0.2 (b)

2.0

2

W ∗/h = 0.0

1

W ∗/h = 0.1 0.0

0.2 ∆x (mm)

0.4

0.6

FIGURE 3.4 Thermopiezoelectric effects on the postbuckling behavior of (FGM=P)S plates with unload edges immovable: (a) load deﬂection; (b) load shortening.

unloaded edges under the same loading conditions, compare Figures 3.3 and 3.7, and Figures 3.4 and 3.8. In contrast, the postbuckling load-carrying capacity of the plate with immovable unloaded edges is larger than that of the plate with movable unloaded edges when the deﬂection W is sufﬁciently large. It is appreciated that in Figures 3.3 through 3.8 W*=h ¼ 0.1 denotes the dimensionless maximum initial geometric imperfection of the plate, and results only for Si3N4=SUS304 hybrid laminated plate under TD case.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 100

Px (kN)

80 60

(P/FGM)S I: N = 0.2 T0 = 300 K, ∆T = 100 K II: N = 1.0 b = 1.0, b/h = 40 III: N = 5.0 (m, n) = (1, 1) 1 3 2

III

II

I

1: VU = VL = −500 V 2: VU = VL = 0 V 3: VU = VL = +500 V

40 Immovable edges 20

W ∗/h = 0.0 W ∗/h = 0.1

0 0.0

0.5

1.0

(a)

1.5

2.0

2.5

W (mm) 100 (P/FGM)S T0 = 300 K, ∆T = 100 K b = 1.0, b/h = 40 (m, n) = (1, 1)

Px (kN)

80 60

II

III

I

1: VU = VL = −500 V

I: N = 0.2 II: N = 1.0 III: N = 5.0

40

2: VU = VL = 0 V 3: VU = VL = +500 V Immovable edges

20

3 0 −0.2 (b)

2

1

W ∗/h = 0.0 W ∗/h = 0.1

−0.1

0.0 ∆x (mm)

0.1

0.2

FIGURE 3.5 Effects of volume fraction index N on the postbuckling behavior of (P=FGM)S plates with unload edges immovable: (a) load deﬂection; (b) load shortening.

3.3

Thermal Postbuckling Behavior of FGM Plates with Piezoelectric Actuators

We now consider thermal buckling problem of these two types of hybrid laminated plate. The temperature ﬁeld is assumed to be a parabolic distribution in the XY-plane of the plate, but uniform through the plate thickness, i.e.,

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Postbuckling of Shear Deformable FGM Plates 150

100

67

III

(FGM/P)S I: N = 0.2 T0 = 300 K, ∆T = 100 K II: N = 1.0 b = 1.0, b/h = 40 III: N= 5.0 (m, n) = (1, 1) 2

Px (kN)

3

II I

1

1: VM = −500 V 2: VM = 0 V 3: VM = +500 V

50

Immovable edges W ∗/h = 0.0 W ∗/h = 0.1

0 0.0

0.5

1.0

1.5

2.0

150

Px (kN)

100

(FGM/P)S T0 = 300 K, ∆T = 100 K b = 1.0, b/h = 40 (m, n) = (1, 1)

III II I

I: N = 0.2 II: N = 1.0 III: N= 5.0

50

2 0 −0.2

1: VM = −500 V 2: VM = 0 V 3: VM = +500 V

Immovable edges

1

W ∗/h = 0.0

3

(b)

2.5

W (mm)

(a)

W ∗/h = 0.1

−0.1

0.0 ∆x (mm)

0.1

0.2

FIGURE 3.6 Effects of volume fraction index N on the postbuckling behavior of (FGM=P)S plates with unload edges immovable: (a) load deﬂection; (b) load shortening.

"

#" # 2X a 2 2Y b 2 T(X, Y) ¼ T1 þ T2 1 1 a b

(3:36)

where T1 is the uniform temperature rise T2 is the temperature gradient All four edges of the plate are assumed to be simply supported with no in-plane displacements. The boundary conditions become

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 250

(P/FGM)S T0 = 300 K b = 1.0, b/h = 40 (m, n) = (1, 1)

Px (kN)

200

I: N = 0.2 II: N = 2.0

150

1

II 2

1: ∆T = 0 K 2: ∆T = 100 K 3: ∆T = 200 K

I

3

100 Movable edges

50

W ∗/h = 0.0 W ∗/h = 0.1

0 (a)

250

200

Px (kN)

1

0

150

3

2

W (mm)

(P/FGM)S T0 = 300 K b = 1.0, b/h = 40 (m, n) = (1, 1)

II

I: N = 0.2 II: N = 2.0

I

II

II

I

I 1: ∆T = 0 K 2: ∆T = 100 K 3: ∆T = 200 K

100

0 −0.4 (b)

Movable edges

1

50

3

W ∗/h = 0.0 W ∗/h = 0.1

2 −0.2

0.0

∆x (mm)

0.2

0.4

0.6

FIGURE 3.7 Effects of temperature rise on the postbuckling behavior of (P=FGM)S plates with unload edges movable: (a) load deﬂection; (b) load shortening.

X ¼ 0, a: W ¼ Cy ¼ 0

(3:37a)

U¼0

(3:37b)

N xy ¼ 0,

Mx ¼ P x ¼ 0

(3:37c)

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Postbuckling of Shear Deformable FGM Plates

69

250

Px (kN)

200

150

(FGM/P)S T0 = 300 K b = 1.0, b/h = 40 (m, n) = (1, 1) 1

2

I: N = 0.2 II: N = 2.0

II I

3

1: ∆T= 0 K 2: ∆T= 100 K 3: ∆T= 200 K

100 Movable edges W ∗/h = 0.0

50

W ∗/h = 0.1

0 0

(a)

250

Px (kN)

200

150

1

3

2

W (mm)

(FGM/P)S T0 = 300 K b = 1.0, b/h = 40 (m, n) = (1, 1)

II

II

I

I

I: N = 0.2 II: N = 2.0

0 − 0.4 (b)

2: ∆T = 100 K 3: ∆T = 200 K 1

3

Movable edges W ∗/h = 0.0 W ∗/h = 0.1

2 −0.2

I 1: ∆T = 0 K

100

50

II

0.0

0.2

0.4

0.6

∆x (mm)

FIGURE 3.8 Effects of temperature rise on the postbuckling behavior of (FGM=P)S plates with unload edges movable: (a) load deﬂection; (b) load shortening.

Y ¼ 0, b: W ¼ Cx ¼ 0

(3:37d)

V¼0

(3:37e)

N xy ¼ 0,

My ¼ Py ¼ 0

(3:37f)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Introducing dimensionless quantities of Equation 2.8, and * D22 * ]1=2 , lT ¼ a0 DT (gT1 , g T2 , gP1 , g P2 ) ¼ ATx , ATy , BPx , BPy a2 =p2 [D11

(3:38)

where a0 is an arbitrary reference value, and a11 ¼ a11 a0 ,

a22 ¼ a22 a0

(3:39)

In Equation 3.39, ATx ( ¼ ATy ) and BEx ( ¼ BEy ) are deﬁned by "

ATx

# DT ¼

ATy "

# tk " X ð Ax k

#

Ay

tk1

X BEx DV ¼ BEy k

ðtk tk1

DT(X, Y)dZ

(3:40a)

k

Bx By

Vk dZ h k p

(3:40b)

where DT ¼ T2 T0 for the in-plane parabolic temperature variation. The nonlinear Equations 3.2 through 3.5 may then be written in dimensionless form as L11 (W) L12 (Cx ) L13 (Cy ) ¼ g 14 b2 L(W þ W*, F) L21 (F)

32 1 lT C1 ¼ g24 b2 L(W þ 2W*, W) 2 p 2

(3:41) (3:42)

L31 (W) þ L32 (Cx ) L33 (Cy ) ¼ 0

(3:43)

L41 (W) L42 (Cx ) þ L43 (Cy ) ¼ 0

(3:44)

where all nondimensional linear operators Lij() and nonlinear operator L() are deﬁned by Equation 2.14. The boundary conditions expressed by Equation 3.37 become x ¼ 0, p: W ¼ Cy ¼ 0

(3:45a)

dx ¼ 0

(3:45b)

F,xy ¼ Mx ¼ Px ¼ 0

(3:45c)

W ¼ Cx ¼ 0

(3:45d)

dy ¼ 0

(3:45e)

y ¼ 0, p:

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Postbuckling of Shear Deformable FGM Plates

71

F,xy ¼ My ¼ Py ¼ 0

(3:45f)

in which 1 dx ¼ 2 2 4p b g24

ðp ðp ( g224 b2 0 0

@2F @2F 1 @W 2 @W @W* g g g 24 5 24 2 2 @y @x 2 @x @x @x

þ g224 gT1 g5 gT2 lT C3 þ g 224 gP1 g5 gP2 DV dx dy 1 dy ¼ 2 2 4p b g24

ðp ðp ( 0 0

(3:46a)

2 2 @2F 1 @W @W* 2@ F 2 @W g g b b g 24 b2 5 @x2 @y2 2 24 @y @y @y

þ (gT2 g5 gT1 ) lT C3 þ ðgP2 g 5 gP1 ÞDV dy dx

(3:46b)

Note that in Equations 3.42 and 3.46, for the in-plane nonuniform parabolic temperature loading case,

C1 ¼ b2 g224 g T1 g 5 gT2 (x=p x2 =p2 ) þ (gT2 g 5 g T1 )(y=p y2 =p2 ), C3 ¼ T1 =T2 þ 16(x=p x2 =p2 )(y=p y2 =p2 ), and for the uniform temperature loading case, C1 ¼ 0,

C3 ¼ 1:0

By using a two-step perturbation technique, we obtain up to fourth-order asymptotic solutions h i W ¼ « A(1) 11 sin mx sin ny h i (3) þ «3 A(3) sin mx sin 3ny þ A sin 3mx sin ny þ O(«5 ) 13 31

(3:47)

2 2 y y5 y6 x5 x5 (0) x C þ C b C þ C F ¼ B(0) 5 5 6 6 00 00 2 120 360p 2 120 360p 2 2 5 6 y y x5 x5 (2) y (2) x 2 þ « B00 C5 þ C5 b00 C6 þ C6 2 120 360p 2 120 360p (2) þ B(2) 20 cos 2mx þ B02 cos 2ny 2 2 y y5 y6 x5 x5 (4) x þ «4 B(4) C þ C b C þ C 5 5 6 6 00 00 2 120 360p 2 120 360p (4) (4) (4) þ B(4) 20 cos 2mx þ B02 cos 2ny þ B22 cos 2mx cos 2ny þ B40 cos 4mx (4) (4) þ B(4) cos 4ny þ B cos 2mx cos 4ny þ B cos 4mx cos 2ny þ O(«5 ) 04 24 42

(3:48)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells h i Cx ¼ « C(1) cos mx sin ny 11 h i (3) 3 þ « C(3) cos mx sin 3ny þ C cos 3mx sin ny þ O(«5 ) 13 31 h i Cy ¼ « D(1) 11 sin mx cos ny h i (3) þ «3 D(3) sin mx cos 3ny þ D sin 3mx cos ny þ O(«5 ) 13 31

(3:49)

(3:50)

Note that for the uniform temperature loading case, it is just necessary to take C5 ¼ C6 ¼ 0 in Equation 3.48, so that the asymptotic solutions have a similar form as expressed by Equations 3.28 through 3.31. Next, upon substitution of Equations 3.47 through 3.50 into the boundary conditions dx ¼ 0 and dy ¼ 0, one has 2 2 (2) 4 2 (4) b2 B(0) 00 þ « b B00 þ « b B00 þ ¼ lT C7

2 (2) 4 (4) b(0) 00 þ « b00 þ « b00 þ ¼ lT C8 (j)

2 1 m2 þ g5 n2 b2 (1 þ 2m) A(1) (3:51a) 11 « 2 2 8 g24 g5

2 1 g5 m2 þ g 224 n2 b2 (1) (1 þ 2m) A « (3:51b) 11 8 g 224 g25

(j)

By adding B00 and b00 (i ¼ 0, 2, 4, . . . ), one has h i (0) 2 2 (2) 4 2 (4) 2 2 (2) 4 (4) 2 2 b2 B(0) 00 þ « b B00 þ « b B00 þ m C9 þ b00 þ « b00 þ « b00 þ n b C10 2 1 4 S11 1 (3:52) Q22 A(1) C11 C44 A(1) ¼ 11 « 11 « þ (1 þ m) 16(1 þ m) 256

g14

From Equations 3.51 and 3.52, the thermal postbuckling equilibrium path can be written as (2) 2 (4) 4 lT ¼ l(0) T þ lT Wm þ lT Wm þ

(3:53)

in which Wm is the dimensionless form of maximum deﬂection, which is assumed to be at the point (x, y) ¼ (p=2m, p=2n). It is noted that l(i)T (i ¼ 0, 2, 4, . . . ) are all functions of temperature and position and given in detail in Appendix G. To obtain numerical results, it is necessary to solve Equation 3.53 by an iterative numerical procedure with the following steps: 1. Begin with W=h ¼ 0. 2. Assume elastic constants and the thermal expansion coefﬁcients are constant, i.e., at T0 ¼ 300 K. The thermal buckling load for the plate of temperature-independent material is obtained. 3. Use the temperature determined in the previous step, the temperature-dependent material properties may be decided from Equations 1.4 and 3.35, and the thermal buckling load is obtained again.

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Postbuckling of Shear Deformable FGM Plates

73

3

∆T/∆Tcr

2

Isotropic plate b = 1.0, b/h = 10 (m, n) = (1, 1) Present W ∗/h = 0.0 W ∗/h = 0.1 Bhimaraddi and Chandrashekhara (1993)

1

W ∗/h = 0.0

0 0.0

W ∗/h = 0.1

0.2

0.4

0.6

0.8

1.0

1.2

W/h FIGURE 3.9 Comparisons of thermal postbuckling load–deﬂection curves for isotropic plates subjected to uniform temperature rise.

4. Repeat step (3) until the thermal buckling temperature converges. 5. Specify the new value of W=h, and repeat steps (2)–(4) until the thermal postbuckling temperature converges. The thermal postbuckling load–deﬂection curves for perfect and imperfect, isotropic square plates (b=h ¼ 10, n ¼ 0.3) subjected to a uniform temperature rise are compared in Figure 3.9 with the analytical solutions of Bhimaraddi and Chandrashekhara (1993). The dimensionless critical temperature l*T ¼ a22 DT 104 for these two theories are identical and l*T ¼ 119.783. In addition, the thermal postbuckling load–deﬂection curves for an isotropic square plate (b=h ¼ 40) subjected to nonuniform parabolic temperature loading with T0=T1 ¼ 1.0 are compared in Figure 3.10 with the ﬁnite difference method solutions of Kamiya and Fukui (1982). These two comparisons show that the results from the present method are in good agreement with the existing results, thus verifying the reliability and accuracy of the present method. Tables 3.7 through 3.12 give thermal buckling loads DTcr (in K) for perfect, (P=FGM)S and (FGM=P)S hybrid laminated plates with different values of the volume fraction index N ( ¼ 0.0, 0.2, 0.5, 1.0, 2.0, and 5.0) subjected to a uniform temperature rise (T2 ¼ 0) and nonuniform parabolic temperature variation with three thermal load ratio T1=T2 ( ¼ 0.0, 0.5, and 1.0). It can be found that the buckling temperature of (P=FGM)S plate is lower than that of (FGM=P)S plate under the same loading condition. As in the case of axial compression, for the hybrid plates with Si3N4=SUS304 substrate, a fully metallic plate (N ¼ 0) has lowest buckling temperature and that the buckling temperature increases as the volume fraction index N increases. It can be seen

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 60 Parabolic temperature field isotropic plate T0/T1 = 1.0

T (⬚C)

40

b =1.0, b/h = 40 (m, n) = (1, 1)

20 Present (HSDPT) Kamiya and Fukui (CPT) (1982) 0 0.0

0.2

0.4

0.6

0.8

1.0

W/h FIGURE 3.10 Comparisons of thermal postbuckling load–deﬂection curves for an isotropic plate subjected to nonuniform parabolic temperature loading.

TABLE 3.7 Comparisons of Buckling Temperature DT (K) for FGM Hybrid Plates with Si3N4=SUS304 Substrate under Uniform Temperature Rise and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) VU( ¼ VL) (in V)

N ¼ 0.0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

500 0

190.8831 188.3028

219.4916 216.8932

250.1407 247.4695

282.0667 279.2786

314.3084 311.3615

345.7490 342.5981

þ500

185.7225

214.2948

244.7983

276.4905

308.4147

339.4472

(P=FGM)S, TD-F 500

172.2518

195.3290

219.4843

244.0116

267.8233

289.8148

0

170.0603

193.1553

217.2865

241.7598

265.4975

287.3994

þ500

167.8666

190.9797

215.0868

239.5061

263.1693

284.9810

(P=FGM)S, TD 500

171.4933

194.4750

218.5124

242.9146

266.5796

288.4171

0

169.4166

192.4294

216.4595

240.8252

264.4368

286.2063

þ500

167.3354

190.3797

214.4023

238.7315

262.2893

283.9898

N ¼ 0.0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

500

229.1825

265.1534

303.8535

344.5095

386.0890

427.2245

0

226.6022

262.5550

301.1823

341.7214

383.1422

424.0736

þ500

224.0218

259.9565

298.5111

338.9334

380.1953

420.9227

(P=FGM)S, TID

VM (FGM=P)S, TID`

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Postbuckling of Shear Deformable FGM Plates

75

TABLE 3.7 (continued) Comparisons of Buckling Temperature DT (K) for FGM Hybrid Plates with Si3N4=SUS304 Substrate under Uniform Temperature Rise and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) VM

N ¼ 0.0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

(FGM=P)S, TD-F 500 0

202.5830 200.4486

230.7437 228.6269

260.2219 258.0835

290.1927 288.0067

319.4639 317.2147

346.4602 344.1400

þ500

198.3126

226.5088

255.9439

285.8196

314.9637

341.8174

(FGM=P)S, TD 500 0

202.4387 200.4283

230.5792 228.6011

260.0317 258.0510

289.9734 287.9671

319.2144 317.1681

346.1820 344.0878

þ500

198.4141

226.6195

256.0670

285.9752

315.1177

341.9890

TABLE 3.8 Comparisons of Buckling Temperature DT (K) for FGM Hybrid Plates with ZrO2=Ti-6Al-4V Substrate under Uniform Temperature Rise and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) VU( ¼ VL) (in V)

N ¼ 0.0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

500

361.0100

313.6715

272.8378

238.9043

209.9199

184.1798

0

352.2897

307.0133

267.6012

234.6535

206.3736

181.1541

þ500

343.5695

300.3551

262.3647

230.4026

202.8273

178.1284

500

339.2903

230.5015

187.5004

161.2948

142.2591

126.8038

0

329.9423

226.2742

184.5547

158.9790

140.3220

125.1165

þ500

320.6644

222.0210

181.5875

156.6469

138.3721

123.4188

500

332.9886

228.5898

186.3837

160.5363

141.7007

126.3721

0

324.7029

224.6504

183.6053

158.3339

139.8469

124.7491

þ500

316.4470

220.6800

180.8029

156.1134

137.9791

123.1148

N ¼ 0.0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

500

398.6698

352.4184

310.5889

274.8633

243.6992

215.4438

0 þ500

389.9496 381.2293

345.7601 339.1019

305.3524 300.1158

270.6125 266.3616

240.1529 236.6066

212.4182 209.3925

500

360.4985

246.6464

202.8977

175.8595

156.4976

140.4670

0 þ500

351.4150 342.4201

242.6464 238.6230

200.1012 197.2875

173.8893 171.6830

154.6699 152.8321

138.8813 137.2873

500

359.3701

246.3116

202.7124

175.7311

156.4056

140.3956

0 þ500

351.2241 343.1091

242.5951 238.8494

200.0737 197.4161

173.8714 171.7710

154.6572 152.8982

138.8722 137.3400

(P=FGM)S, TID

(P=FGM)S, TD-F

(P=FGM)S, TD

VM (FGM=P)S, TID

(FGM=P)S, TD-F

(FGM=P)S, TD

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TABLE 3.9 Comparisons of Buckling Temperature DT (K) for (P=FGM)S Plates with Si3N4=SUS304 Substrate under Nonuniform Parabolic Temperature Variation and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) T1=T2

VU( ¼ VL) (in V)

N ¼ 0.0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

334.1945 329.6768

384.2758 379.7267

493.8177 488.9366

550.2584 545.0994

605.2963 599.7801

(P=FGM)S, TID 0.0

500 0 þ500

325.1593

375.1775

484.0555

539.9403

594.2639

0.5

500

178.2000

204.9059

263.3194

293.4167

322.7661

0

175.7911

202.4802

260.7166

290.6657

319.8246

þ500

173.3822

200.0544

258.1139

287.9147

316.8832

500

121.4908

139.6983

179.5235

200.0434

220.0532

0

119.8485

138.0446

177.7491

198.1678

218.0478

þ500

118.2062

136.3908

175.9746

196.2923

216.0424

(P=FGM)S, TD-F 0.0 500

284.4204

321.6889

399.2022

435.4759

466.3745

0 þ500

280.8110 277.2018

318.0988 314.5095

395.4951 391.7880

431.6948 427.9120

462.5440 458.7091

500

161.7491

183.5069

229.4070

252.0241

272.9636

0

159.6876

181.4618

227.2878

249.8317

270.6824

þ500

157.6241

179.4149

225.1668

247.9120

268.3984

500

113.3601

128.9681

162.1571

178.6939

194.3458

0

111.8991

127.5158

160.6434

177.1206

192.6956

þ500

110.4367

126.0622

159.1284

175.5458

191.0437

(P=FGM)S, TD 0.0 500

462.8696

1.0

0.5

1.0

282.4131

319.4363

396.3263

432.2831

0

279.1118

316.1897

393.0523

428.9790

459.5585

þ500

275.8039

312.9370

389.7715

425.6670

456.2370

0.5

500 0

161.0838 159.1237

182.7577 180.8258

228.5001 226.5236

250.9276 248.8976

271.7272 269.6281

þ500

157.1594

178.8899

224.4809

246.8633

267.5240

1.0

500

113.0288

128.5932

161.6644

178.1279

193.6974

0

111.6182

127.1975

160.2241

176.6385

192.1429

þ500

110.2051

125.7995

158.7813

175.1463

190.5851

that an increase is about þ69% for the (P=FGM)S plate, and about þ72% for the (FGM=P)S one under uniform temperature rise, and about þ65% for the (P=FGM)S plate, and about þ68% for the (FGM=P)S one under nonuniform parabolic temperature variation with T1=T2 ¼ 0.0, from N ¼ 0 to N ¼ 5 under TD case. It can also be seen that the buckling temperature decreases when the temperature dependency is put into consideration. The percentage decrease is about 15% for the (P=FGM)S plate and about 17% for the (FGM=P)S one from TID case to TD case under uniform temperature

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Postbuckling of Shear Deformable FGM Plates

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TABLE 3.10 Comparisons of Buckling Temperature DT (K) for (FGM=P)S Plates with Si3N4=SUS304 Substrate under Nonuniform Parabolic Temperature Variation and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) T1=T2

VM (in V)

N ¼ 0.0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

401.2482 396.7307

464.2183 459.6692

603.1371 598.2560

675.9244 670.7654

747.9340 742.4178

(FGM=P)S, TID 0.0

500 0 þ500

392.2130

455.1200

593.3749

665.6063

736.9016

0.5

500

213.9545

247.5333

321.6120

360.4261

398.8257 395.8842

1.0

0

211.5456

245.1076

319.0093

357.6752

þ500

209.1367

242.6819

316.4065

354.9242

392.9428

500

145.8671

168.7604

219.2658

245.7285

271.9086

0

144.2248

176.1066

217.4913

243.8530

269.9032

þ500

142.5825

165.4528

215.7168

241.9774

267.8978

(FGM=P)S, TD-F 0.0 500

330.7502

376.1384

471.0853

515.2603

552.2033

0 þ500

327.2496 323.7512

372.6424 369.1491

467.4599 463.8368

511.5790 507.8985

548.5238 544.8420

500

190.4493

216.9529

273.1703

300.9339

326.7027

0

188.4398

214.9614

271.1117

298.8130

324.5098

þ500

186.4289

212.9685

269.0516

296.6906

322.3148

500

134.1980

153.3095

194.0825

214.5766

234.1622

0.5

1.0

0

132.7667

151.8893

192.6086

213.0498

232.5667

þ500

131.3342

150.4680

191.1337

211.5216

230.9696

(FGM=P)S, TD 0.0 500

330.4118

375.7495

470.5706

514.6824

551.5844

0

327.2365

372.6227

467.4229

511.5351

548.4777

þ500

324.0568

369.4919

464.2708

508.3820

545.3630

0.5

500 0

190.3281 188.4281

216.8136 214.9458

272.9858 271.0864

300.7233 298.7827

326.4663 324.4752

þ500

186.5247

213.0748

269.1836

296.8385

322.4798

1.0

500

134.1359

153.2376

193.9849

214.4635

234.0341

0

132.7595

151.8797

192.5931

213.0307

232.5446

þ500

131.3808

150.5197

191.1989

211.5955

231.0524

rise and the same volume fraction distribution N ¼ 2. Also the thermal buckling loads under TD-F and TD cases are very close under the same volume fraction distribution and the same thermal loading condition. In contrast, it is seen that the buckling temperature of hybrid plates with ZrO2=Ti-6Al-4V substrate is decreased as the volume fraction index N increases. It can also be seen that the control voltage has a small effect on the thermal buckling loads for these hybrid laminated plates.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 3.11 Comparisons of Buckling Temperature DT (K) for (P=FGM)S Plates with ZrO2=Ti-6Al-4V Substrate under Nonuniform Parabolic Temperature Variation and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) T1=T2

VU( ¼ VL) (in V)

N ¼ 0.0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

632.1551 616.8853

549.2456 537.5869

418.3071 410.8641

367.5503 361.3410

322.4765 317.1789

(P=FGM)S, TID 0.0

500 0 þ500

601.6155

525.9282

403.4211

355.1317

311.8813

0.5

500

337.0529

292.8512

223.0412

195.9794

171.9472

0

328.9113

286.6349

219.0726

192.6686

169.1224

þ500

320.7697

280.4187

215.1040

189.3578

166.2977

500

229.7849

199.6515

152.0597

133.6106

117.2267

0

224.2344

195.4136

149.3541

131.3534

115.3009

þ500

218.6839

191.1756

146.6485

129.0962

113.3752

(P=FGM)S, TD-F 0.0 500

598.0353

343.6654

235.8257

208.3855

186.4033

0 þ500

577.4709 557.6385

337.7048 331.6966

232.6350 229.4191

205.7157 203.0257

184.0908 181.7600

500

317.2877

218.7738

153.5554

135.3958

120.7423

0

308.6649

214.7361

151.3301

133.5351

119.1132

þ500

300.0934

210.6746

149.0898

131.6627

117.4751

500

218.7285

161.5482

115.1498

101.4113

90.0399

0

213.0890

158.4637

113.4055

99.9548

88.7771

þ500

207.4749

155.3648

111.6517

98.4907

87.5082

(P=FGM)S, TD 0.0 500

185.5407

1.0

0.5

1.0

574.8693

339.7483

234.0814

207.3249

0

558.7789

334.3736

231.4190

204.8110

183.3573

þ500

542.9512

328.9448

228.4113

202.2762

181.1540

0.5

500 0

311.8759 304.1632

217.0459 213.2708

152.8709 150.7486

134.8918 133.1068

120.3626 118.7900

þ500

296.4737

209.4670

148.6098

131.3090

117.2077

1.0

500

216.2538

160.5520

114.7444

101.1136

89.8113

0

211.0199

157.6210

113.0615

99.7021

88.5831

þ500

205.7910

154.6718

111.3681

98.2823

87.3481

Figure 3.11 shows the effect of material properties on the thermal postbuckling load–deﬂection curves for (P=FGM)S and (FGM=P)S hybrid laminated plates (N ¼ 2.0) subjected to a uniform temperature rise (T2 ¼ 0) and nonuniform parabolic temperature variation (T1=T2 ¼ 0.0), respectively, and with three sets of electrical loading, VU ¼ VL (or VM) ¼ 500, 0, þ500 V, and under two cases of thermoelastic material properties, i.e., TID and TD. It can be seen that the thermal postbuckling equilibrium path becomes lower

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Postbuckling of Shear Deformable FGM Plates

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TABLE 3.12 Comparisons of Buckling Temperature DT (K) for (FGM=P)S Plates with ZrO2= Ti-6Al-4V Substrate under Nonuniform Parabolic Temperature Distribution and Three Sets of Electrical Loading Conditions (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) T1=T2

VM (in V)

N ¼ 0.0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

698.1001 682.8303

617.0921 605.4334

481.2691 473.8261

426.6948 420.4855

377.2160 371.9184

(FGM=P)S, TID 0.0

500 0 þ500

667.5605

593.7747

466.3831

414.2762

366.6208

0.5

500

372.2135

329.0262

256.6125

227.5155

201.1348 198.3100

1.0

0

364.0719

322.8099

252.6440

224.2047

þ500

355.9304

316.5936

248.6754

220.8939

195.4853

500

253.7555

224.3138

174.9472

155.1106

137.1256

0

248.2050

220.0759

172.2415

152.8534

135.1999

þ500

242.6546

215.8379

169.5359

150.5962

133.2741

(FGM=P)S, TD-F 0.0 500

614.2082

362.4850

253.1466

225.7529

203.2276

0 þ500

595.3040 576.9683

356.8670 351.2095

250.2042 247.2384

223.2697 220.7703

201.0907 198.9386

500

337.9622

234.5947

167.8398

149.1455

133.6344

0

329.5637

230.7643

165.7353

147.3920

132.1239

þ500

321.2380

226.9125

163.6189

145.6291

130.6051

500

235.5359

175.0072

127.0871

112.8256

100.9174

0.5

1.0

0

229.9974

172.0518

125.4258

111.4420

99.7203

þ500

224.4839

169.0833

123.7565

110.0521

98.4001

(FGM=P)S, TD 0.0 500

610.6959

361.8958

252.9243

225.5958

203.0985

0

595.0753

356.8408

250.1978

223.2656

201.0885

þ500

579.6954

351.7393

247.4456

220.9188

199.0620

0.5

500 0

337.0280 329.4402

234.3057 230.7321

167.7308 165.7247

149.0652 147.3848

133.5706 132.1188

þ500

321.8768

227.1318

163.7057

145.6942

130.6580

1.0

500

235.0930

174.8349

127.0197

112.7761

100.8794

0

229.9281

172.0281

125.4175

111.4363

99.7164

þ500

224.7690

169.2046

123.8067

110.0897

98.4306

when the temperature-dependent properties are taken into account. It can also be seen that the FGM hybrid laminated plate under a uniform temperature rise has a lower postbuckling temperature than does a plate under a nonuniform parabolic temperature variation. It is found that minus control voltages increase the buckling temperature and decrease the postbuckled deﬂection, whereas the plus control voltages decrease the buckling temperature and induce more large postbuckled deﬂections. Since the effect of the

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 1500

I: uniform temperature (T2 = 0) II: parabolic temperature (T1/T2 = 0.0) (P/FGM)S, N = 2.0 T0 = 300 K b = 1.0, b/h = 20 (m, n) = (1, 1)

∆T (K)

1000

II: T-ID (1, 2, 3)

II: T-D (1, 2, 3) 1: VU = VL = −500 V I: T-ID (1, 2, 3) 2: VU = VL = 0 V 3: VU = VL = +500 V I: T-D (1, 2, 3)

500

W ∗/h = 0.0

(a)

0 0.0

W ∗/h = 0.05

0.5

1.0

1.5 W (mm)

1500 I: uniform temperature (T2 = 0) II: parabolic temperature (T1/T2 = 0.0)

∆T (K)

1000

(P/FGM)S, N = 2.0 T0 = 300 K b = 1.0, b/h = 20 (m, n) = (1, 1)

2.0

2.5

II: T-ID (1, 2, 3)

II: T-D (1, 2, 3) 1: VU = VL = −500 V I: T-ID (1, 2, 3) 2: VU = VL = 0 V 3: VU = VL = +500 V

500

I: T-D (1, 2, 3)

W ∗/h = 0.0 W ∗/h = 0.05 0 0.0 (b)

0.5

1.0

1.5

2.0

2.5

W (mm)

FIGURE 3.11 Effect of material properties on the thermal postbuckling behavior of FGM hybrid laminated plates under in-plane temperature variation: (a) (P=FGM)S plate; (b) (FGM=P)S plate.

control voltages is small, these three curves, referred to as 1, 2, and 3 in the ﬁgure, are very close. Figure 3.12 shows the effect of the volume fraction index N ( ¼ 0.2, 1.0, and 5.0) on the thermal postbuckling behavior of (P=FGM)S and (FGM=P)S hybrid laminated plates subjected to a nonuniform parabolic temperature variation with T1=T2 ¼ 0.5 and three sets of electrical loading. It can be seen that the increase of the volume fraction index N yields an increase of the buckling temperature and thermal postbuckling strength. Figure 3.13 shows the effect of different values of the thermal load ratio T1=T2 ( ¼ 0.0, 0.5, 1.0) on the postbuckling behavior of (P=FGM)S and

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Postbuckling of Shear Deformable FGM Plates

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1000 Parabolic temperature I: N = 0.2 (P/FGM)S II: N = 1.0 T0 = 300 K, T1/T2 = 0.5 III: N = 5.0 b = 1.0, b/h = 20 (m, n) = (1, 1)

∆T (K)

800 600

III (1, 2, 3) 1: VU = VL = −500 V II (1, 2, 3) 2: VU = VL = 0 V 3: VU = VL = +500 V I (1, 2, 3)

400 200

W ∗/h = 0.0

0 0.0

W ∗/h = 0.05

0.5

1.0

(a)

1.5

2.0

2.5

W (mm) 1000 Parabolic temperature (FGM/P)S T0 = 300 K, T1/T2 = 0.5 b = 1.0, b/h = 20 (m, n) = (1, 1)

800

∆T (K)

600

I: N = 0.2 II: N = 1.0 III: N = 5.0

III (1, 2, 3) II (1, 2, 3)

1: VM = −500 V

I (1, 2, 3) 2: VM = 0 V 3: VM = +500 V

400

200

W ∗/h = 0.0 W ∗/h = 0.05

0 (b)

0

1

W (mm)

2

3

FIGURE 3.12 Effect of volume fraction index N on the thermal postbuckling behavior of FGM hybrid laminated plates under nonuniform parabolic temperature variation: (a) (P=FGM)S plate; (b) (FGM=P)S palte.

(FGM=P)S hybrid laminated plates subjected to a nonuniform parabolic temperature variation and three sets of electrical loading. It can be found that the postbuckling strength is decreased by increasing T1=T2, when the volume fraction index N ¼ 2.0. As mentioned in Section 3.2, in Figures 3.11 through 3.13, W*=h ¼ 0.05 denotes the dimensionless maximum initial geometric imperfection of the plate.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 1200

Parabolic temperature I: T /T = 0.0 1 2 (P/FGM)S II: T1/T2 = 0.5 T0 = 300 K, N = 2.0 III: T1/T2 = 1.0 b = 1.0, b/h = 20 (m, n) = (1, 1)

800

I (1, 2, 3)

∆T (K)

II (1, 2, 3) 1: VU = VL = −500 V 2: VU = VL = 0 V 3: VU = VL = +500 V

III (1, 2, 3)

400

W ∗/h = 0.0

(a)

0 0.0

W ∗/h = 0.05 0.5

1.0

1.5

2.0

2.5

W (mm)

1200 Parabolic temperature I: T /T = 0.0 1 2 (FGM/P)S II: T1/T2 = 0.5 T0 = 300 K, N = 2.0 III: T1/T2 = 1.0 b = 1.0, b/h = 20 (m, n) = (1, 1)

II (1, 2, 3) 1: VM = −500 V 2: VM = 0 V

∆T (K)

800

I (1, 2, 3)

III (1, 2, 3)

3: VM = +500 V

400

W ∗/h = 0.0 0 (b)

W ∗/h = 0.05 0.0

0.5

1.0

1.5

2.0

2.5

W (mm)

FIGURE 3.13 Effect of thermal load ratio T1=T2 on the postbuckling of FGM hybrid laminated plates under nonuniform parabolic temperature variation: (a) (P=FGM)S plate; (b) (FGM=P)S plate.

3.4

Postbuckling of Sandwich Plates with FGM Face Sheets in Thermal Environments

Finally, we consider a rectangular sandwich plate which consists of one homogeneous substrate and two face sheets made of functionally graded materials and is midplane symmetric, as shown in Figure 3.14. The length, width, and total thickness of the sandwich plate are a, b, and h. The thickness

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Postbuckling of Shear Deformable FGM Plates

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a t0 t1

b

h

hF

FGM

hH

Homogeneous

X t2 t3

FGM Y Z FIGURE 3.14 Conﬁguration of a sandwich plate.

of each FGM face sheet is hF, while the thickness of the homogeneous substrate is hH. The FGM face sheet is made from a mixture of ceramics and metals, the mixing ratio of which is varied continuously and smoothly in the Z-direction. It is assumed that the effective Young’s modulus Ef, thermal expansion coefﬁcient af, and thermal conductivity kf of FGM face sheets are functions of temperature, so that Ef, af, and kf are both temperature- and position-dependent. The Poisson ratio nf depends weakly on temperature change and is assumed to be a constant. We assume the volume fraction Vm follows a simple power law. According to rule of mixture, we have

Z t0 N þ Ec (T) t1 t0 Z t0 N þ ac (T) af (Z, T) ¼ [am (T) ac (T)] t1 t0 Z t0 N þ kc (T) kf (Z, T) ¼ [km (T) kc (T)] t1 t0 Ef (Z, T) ¼ [Em (T) Ec (T)]

(3:54a) (3:54b) (3:54c)

It is evident that when Z ¼ t0, Ef ¼ Ec, af ¼ ac and kf ¼ kc, and when Z ¼ t1, Ef ¼ Em, af ¼ am, and kf ¼ km. The temperature ﬁeld is assumed to be uniform over the plate surface but varying along the thickness direction due to heat conduction which can be determined by solving a steady-state heat transfer equation d dT k ¼0 dZ dZ

(3:55)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

where 8 < kf k ¼ kH : kf 8 < TF T ¼ TH : ~F T

(t0 Z < t1 ) (t1 Z t2 ) (t2 < Z t3 )

(3:56a)

(t0 Z t1 ) (t1 Z t2 ) (t2 Z t3 )

(3:56b)

where kH is the thermal conductivity of the homogeneous substrate. Equation 3.55 is solved by imposing the boundary conditions T ¼ TU at Z ¼ t0 and T ¼ TL at Z ¼ t3, and the continuity conditions ~ F (t2 ) ¼ Tm2 TF (t1 ) ¼ TH (t1 ) ¼ Tm1 , TH (t2 ) ¼ T ~F dTF dTH dTH dT km ¼ k , k ¼ k H H m dZ Z¼t1 dZ Z¼t1 dZ Z¼t2 dZ Z¼t2

(3:57a) (3:57b)

where TU and TL are the temperatures at top and bottom surfaces of the plate Tm1 and Tm2 are the temperatures at Z ¼ t1 and t2 interfaces, respectively The solutions of Equations 3.55 through 3.57, by means of polynomial series, are TF ¼ TU þ (Tm1 TU )h(Z) TH ¼

(3:58a)

1 ½(Tm1 t2 Tm2 t1 ) þ (Tm2 Tm1 )Z hH

(3:58b)

~ F (Z) ¼ TL þ (Tm2 TL )~ T h(Z)

(3:58c)

in which "

kmc Z t0 Nþ1 k2mc Z t0 2Nþ1 þ (N þ 1)kc t1 t0 (2N þ 1)k2c t1 t0 # k3mc Z t0 3Nþ1 k4mc Z t0 4Nþ1 k5mc Z t0 5Nþ1 þ (3N þ 1)k3c t1 t0 (4N þ 1)k4c t1 t0 (5N þ 1)k5c t1 t0

h(Z) ¼

1 C

Z t0 t1 t0

(3:59a) "

Nþ1

2Nþ1

Z t3 kmc Z t3 k2mc Z t3 þ t2 t3 (N þ 1)kc t2 t3 (2N þ 1)k2c t2 t3 # k3mc Z t3 3Nþ1 k4mc Z t3 4Nþ1 k5mc Z t3 5Nþ1 þ (3N þ 1)k3c t2 t3 (4N þ 1)k4c t2 t3 (5N þ 1)k5c t2 t3

h ~(Z) ¼

1 C

(3:59b)

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Postbuckling of Shear Deformable FGM Plates C¼1

85

kmc k2mc k3mc k4mc þ þ (N þ 1)kc (2N þ 1)k2c (3N þ 1)k3c (4N þ 1)k4c

k5mc (5N þ 1)k5c G¼1

(3:59c) kmc k2mc k3mc k4mc k5mc þ 2 3 þ 4 5 kc kc kc kc kc

(3:59d)

where kmc ¼ km kc, and ((kH =hH ) þ (km G=hF C))TU þ (kH =hH )TL , (km G=hF C) þ (2kH =hH ) ((kH =hH ) þ (km G=hF C))TL þ (kH =hH )TU ¼ (km G=hF C) þ (2kH =hH )

Tm1 ¼ Tm2

(3:60)

The plate is assumed to be geometrically imperfect, and is subjected to a compressive edge load in the X-direction and=or heat conduction. Two cases of compressive postbuckling under thermal environments and of thermal postbuckling due to heat conduction are considered. Since the heat conduction is considered in the present case, the general von Kármán-type equations (Equations 1.33 through 1.36) can be written in the simple form as ~12 (Cx ) L ~13 (Cy )þ L ~14 (F) L ~15 (N T ) L ~16 (MT ) ¼ L(W ~ þ W*,F) (3:61) ~11 (W) L L ~ ~21 (F) þ L ~22 (Cx ) þ L ~23 (Cy ) L ~24 (W) L ~25 (N T ) ¼ 1 L(W L þ 2W*, W) (3:62) 2 ~31 (W) þ L ~32 (Cx ) L ~33 (Cy ) þ L ~34 (F) L ~35 (N T ) L ~36 (ST ) ¼ 0 L

(3:63)

~41 (W) L ~42 (Cx ) þ L ~43 (Cy ) þ L ~44 (F) L ~45 (N T ) L ~46 (ST ) ¼ 0 L

(3:64)

~ij() and nonlinear operator L() ~ are deﬁned by where all linear operators L Equation 1.37. The forces and moments caused by elevated temperature are deﬁned by Equations 3.8 and 3.9. All four edges of the plate are assumed to be simply supported. Depending upon the in-plane behavior at the edges, two cases, case 1 (for the compressive buckling problem) and case 2 (for the thermal buckling problem), will be considered. They are Case 1: The edges are simply supported and freely movable in the in-plane directions. In addition, the plate is subjected to uniaxial compressive edge loads. Case 2: All four edges are simply supported with no in-plane displacements, i.e., prevented from moving in the X- and Y-directions.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells For both cases the associated boundary conditions can be expressed by

X ¼ 0, a: W ¼ Cy ¼ 0

(3:65a)

Mx ¼ P x ¼ 0

(3:65b)

ðb N x dY þ P ¼ 0 (for compressive buckling problem)

(3:65c)

0

U ¼ 0 (for thermal buckling problem)

(3:65d)

W ¼ Cx ¼ 0

(3:65e)

My ¼ P y ¼ 0

(3:65f)

Y ¼ 0, b:

ða N y dX ¼ 0 (for compressive buckling problem)

(3:65g)

0

V ¼ 0 (for thermal buckling problem)

(3:65h)

where P is a compressive edge load in the X-direction Mx and My are the bending moments Px and Py are the higher order moments, deﬁned by Equation 3.12 The condition expressing the immovability condition, U ¼ 0 (on X ¼ 0, a) and V ¼ 0 (on Y ¼ 0, b), is fulﬁlled on the average sense as ðb ða 0 0

@U dX dY ¼ 0, @X

ða ðb 0 0

@V dY dX ¼ 0 @Y

(3:66)

This condition in conjunction with Equation 3.67 below provides thermally induced compressive stresses acting on the edges X ¼ 0, a and Y ¼ 0, b. The average end-shortening relationships are Dx 1 ¼ a ab

ðb ða 0 0

¼

1 ab

@U dX dY @X

ðb ða (" * A11 0 0

@Cy @2 F @2F 4 @Cx 4 * * * * * þ B12 þ A12 þ B11 E11 E12 2 2 2 2 @X @Y @Y @X 3h 3h

) 2 # 4 @2W @2W 1 @W @W @W* T T * * * N x þ A12 * Ny 2 E11 þ E12 A11 dX dY 3h @X2 @Y2 2 @X @X @X

(3:67a)

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Postbuckling of Shear Deformable FGM Plates Dy 1 ¼ b ab

ða ðb 0 0

87

@V dY dX @Y

ða ðb (" @Cy 1 @2F @2F 4 @Cx 4 * *12 *21 *21 *22 *22 þ B ¼ A22 þ A þ B E E @X @Y ab @X2 @Y2 3h2 3h2 0 0

) 2 # 4 @2 W @2W 1 @W @W @W* T T * * * * 2 E21 þ E22 A12 N x þ A22 N y dY dX 3h @X2 @Y2 2 @Y @Y @Y

(3:67b)

where Dx and Dy are plate end-shortening displacements in the X- and Y-directions. Introducing dimensionless quantities of Equation 2.8, and * D22 * ]1=2 (g T1 , g T2 ) ¼ ATx , ATy a2 =p2 [D11 * (gT3 , g T4 , gT6 , gT7 ) ¼ DTx , DTy , FTx , FTy a2 =p2 h2 D11 * D22 * ]1=2 , lx ¼ Pb=4p2 [D11

(3:68)

lT ¼ a0 DT

where a0 is an arbitrary reference value, deﬁned by Equation 3.39. Also we let "

ATx

DTx

FTx

ATy

DTy

FTy

# DT ¼

# tk " X ð Ax k

tk1

Ay

(1, Z, Z3 )DT(Z)dZ

(3:69)

k

where DT is a constant and is deﬁned by DT ¼ TU T0 for the heat conduction. The nonlinear equations (Equations 3.61 through 3.64) may then be written in dimensionless form as L11 (W) L12 (Cx ) L13 (Cy ) þ g 14 L14 (F) L16 (MT ) ¼ g14 b2 L(W þ W*,F) (3:70) L21 (F) þ g24 L22 (Cx ) þ g24 L23 (Cy ) g 24 L24 (W) 1 ¼ g 24 b2 L(W þ 2W*, W) 2

(3:71)

L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) L36 (ST ) ¼ 0

(3:72)

L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) L46 (ST ) ¼ 0

(3:73)

where all nondimensional linear operators Lij() and nonlinear operator L() are deﬁned by Equation 2.14.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells The boundary conditions expressed by Equation 3.65 become

x ¼ 0, p:

1 p

ðp b2 0

W ¼ Cy ¼ 0

(3:74a)

Mx ¼ P x ¼ 0

(3:74b)

@2F dy þ 4lx b2 ¼ 0 (for compressive buckling problem) @y2

(3:74c)

dx ¼ 0 (for thermal buckling problem)

(3:74d)

W ¼ Cx ¼ 0

(3:74e)

My ¼ P y ¼ 0

(3:74f)

y ¼ 0, p:

ðp 0

@2F dx ¼ 0 (for compressive buckling problem) @x2 dy ¼ 0 (for thermal buckling problem)

(3:74g) (3:74h)

and the unit end-shortening relationships become ðp ðp @Cy 1 @2F @2F @Cx þ g233 b dx ¼ 2 2 g224 b2 2 g 5 2 þ g24 g511 @y @x @x @y 4p b g 24 0 0 2 @2W 1 @W 2 @W @W* 2@ W g g24 g611 þ g b g24 244 24 2 2 @x @y 2 @x @x @x 2

(3:75a) þ g24 g T1 g 5 gT2 lT dx dy ðp ðp 2 2 @Cy 1 @ F @Cx 2@ F þ g g b þ g g b 5 24 220 522 @x @y @x2 @y2 4p2 b2 g 24 0 0 2 2 @2W 1 2@ W 2 @W g24 b þ g 622 b g 24 g 240 @x2 @y2 2 @y @W @W* þ ðgT2 g5 gT1 ÞlT dy dx g 24 b2 (3:75b) @y @y

dy ¼

Since the material properties are assumed to be functions of T and Z, and the temperature is also assumed to be function of Z, even for the midplane symmetric plate the coupling between transverse bending and in-plane

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Postbuckling of Shear Deformable FGM Plates

89

stretching which is given in terms of Bij* and Eij* (i, j ¼ 1, 2, 6) is still existed when the plate is subjected to heat conduction. In contrast, when the plate is subjected to a uniform temperature rise, all Bij* and Eij* are zero-valued. Hence, for the heat conduction case, a nonlinear static problem must be ﬁrst solved to determine the prebuckling deﬂection Wi caused by temperature ﬁeld (Shen 2007a), and then replaced W* with WT* ¼ W * þ Wi in Equations 3.70 and 3.71. Note that for uniform temperature loading case, that is, TU ¼ TL, Wi ¼ 0 and WT* ¼ W * . By using a two-step perturbation technique, the asymptotic solutions are obtained in the same form of Equations 3.28 through 3.31. Upon substitution of these solutions into the boundary conditions (Equations 3.74c and 3.75a), or into the boundary conditions (Equations 3.74d and 3.74h), the compressive postbuckling equilibrium path can be written as (2) 2 (4) 4 lp ¼ l(0) p þ lp Wm þ lp Wm þ

(3:76a)

(2) 2 (4) 4 dx ¼ d(0) x þ dx W m þ dx W m þ

(3:76b)

and the thermal postbuckling equilibrium path can be written as (2) 2 (4) 4 lT ¼ l(0) T þ lT Wm þ lT Wm þ

(3:77)

in which Wm is the dimensionless form of maximum deﬂection, and (i) (i) l(i) p , dp , and lT (i ¼ 0, 2, 4, . . . ) are all functions of temperature and position and given in detail in Appendix H. For numerical illustrations, two sets of material mixture, as shown in Section 3.2, for FGM face sheets are considered. The material properties are assumed to be nonlinear function of temperature of Equation 1.4, and typical values, in the present case, can be found in Tables 1.1 through 1.3. The same metal is selected for the homogeneous substrate, and the material properties of which are also assumed to be nonlinear function of temperature of Equation 1.4. Poisson’s ratio is assumed to be a constant, that is, nf ¼ 0.28 for (Si3N4=SUS304) face sheets and nH ¼ 0.3 for (SUS304) substrate, also nf ¼ nH ¼ 0.29 for both (ZrO2=Ti-6Al-4V) face sheets and (Ti-6Al-4V) substrate. For these examples, the plate geometric parameter a=b ¼ 1, b=h ¼ 20, and the thickness of the FGM face sheet hF ¼ 1 mm whereas the thickness of the homogeneous substrate is taken to be hH ¼ 4, 6, and 8 mm, so that the substrate-to-face sheet thickness ratio hH=hF ¼ 4, 6, and 8, respectively. It should be appreciated that in all ﬁgures W*=h ¼ 0.1 (for the compressive postbuckling) or 0.05 (for the thermal postbuckling) denotes the dimensionless maximum initial geometric imperfection of the plate. Tables 3.13 and 3.14 present the compressive buckling loads Pcr (in kN) for perfect, sandwich plates with different values of the substrate-to-face sheet thickness ratio hH=hF ( ¼ 4, 6, and 8) and with different values of the volume fraction index N ( ¼ 0.0, 0.2, 0.5, 1.0, 2.0, and 5.0) of FGM face sheets subjected to uniaxial compression under three sets of uniform

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 3.13 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, (FGM=SUS304=FGM) Sandwich Plates under Uniform Temperature Rise (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) hH=hF

DT (in K)

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

8.0 6.0

3681.317 2353.399

3859.934 2492.362

4034.281 2627.142

4203.058 2756.482

4364.358 2878.550

4515.310 2990.619

4.0

1321.721

1421.209

1516.661

1606.887

1690.181

1764.056

TID, (m, n) ¼ (1, 1)

TD, (m, n) ¼ (1, 1) 8.0 100

6.0

4.0

3628.076

3801.115

3970.019

4133.529

4289.799

4436.045

200

3528.290

3701.026

3869.631

4032.846

4188.832

4334.807

300

3381.958

3559.376

3732.538

3900.154

4060.333

4210.224

100

2319.363

2453.986

2584.559

2709.864

2828.126

2936.704

200

2255.571

2389.958

2520.300

2645.377

2763.422

2871.797

300

2162.024

2300.053

2433.914

2562.358

2683.572

2794.848

100

1302.606

1398.987

1491.460

1578.872

1659.571

1731.144

200 300

1266.779 1214.241

1362.991 1313.058

1455.298 1407.855

1542.551 1497.453

1623.099 1580.159

1694.538 1663.508

TABLE 3.14 Comparisons of Buckling Loads Pcr (kN) for Uniaxial Compressed, (FGM=Ti-6Al-4V=FGM) Sandwich Plates under Uniform Temperature Rise (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) hH=hF

DT (in K)

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

8.0

1872.292

1970.214

2065.789

2158.303

2246.714

2329.446

6.0

1198.267

1274.447

1348.327

1419.219

1486.120

1547.536

TID, (m, n) ¼ (1, 1)

4.0

674.0251

728.5607

780.8779

830.3261

875.9709

916.4501

TD, (m, n) ¼ (1, 1) 8.0

6.0

4.0

100

1772.731

1854.270

1933.864

2010.920

2084.566

2153.490

200

1673.170

1744.246

1813.632

1880.810

1945.019

2005.116

300 100

1573.609 1134.548

1639.296 1197.983

1703.425 1259.513

1765.512 1318.563

1824.857 1374.296

1880.401 1425.467

200

1070.829

1126.125

1179.764

1231.246

1279.841

1324.460

300

1007.110

1058.214

1107.789

1155.370

1200.284

1241.523

100

638.1832

683.5961

727.1699

768.3617

806.3907

840.1191

200

602.3412

641.9277

679.9152

715.8294

748.9890

778.4004

300

566.4993

603.0850

638.1940

617.3875

702.0355

729.2194

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Postbuckling of Shear Deformable FGM Plates

91

temperature rise (DT ¼ 100, 200, 300 K). Two kinds of sandwich plates, (FGM=SUS304=FGM) and (FGM=Ti-6Al-4V=FGM), are considered. It can be found that the buckling load of (FGM=Ti-6Al-4V=FGM) sandwich plate is lower than that of (FGM=SUS304=FGM) one. It can be seen that a fully metallic plate (N ¼ 0) has lowest buckling load and that the buckling load increases as the volume fraction index N increases. It is found that the increase is about þ36% for the (FGM=SUS304=FGM) sandwich plate, and about þ29% for the (FGM=Ti-6Al-4V=FGM) one with hH=hF ¼ 4, from N ¼ 0 to N ¼ 5, under temperature change DT ¼ 300 K. It can also be seen that the temperature reduces the buckling load when the temperature dependency is put into consideration. The percentage decrease is about 6.5% for the (FGM=SUS304=FGM) sandwich plate and about 20% for the (FGM= Ti-6Al-4V=FGM) one with hH=hF ¼ 4 from temperature changes from DT ¼ 100 K to DT ¼ 300 K under the same volume fraction distribution N ¼ 2. Then Tables 3.15 and 3.16 present thermal buckling temperature DTcr (in K) for the same two kinds of sandwich plates with different values of the volume fraction index N ( ¼ 0.0, 0.2, 0.5, 1.0, 2.0, and 5.0) subjected to a uniform temperature rise. Note that, for the thermal buckling problem, it is necessary to solve Equation 3.77 by an iterative numerical procedure, as previously shown in Section 3.3. The results conﬁrm that the buckling temperature is reduced when the TD properties are taken into account. Now the buckling TABLE 3.15 Comparisons of Buckling Temperature DT (K) for (FGM=SUS304=FGM) Sandwich Plates under Uniform Temperature Rise (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) hH=hF

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

TID

503.4949

513.6594

524.5022

535.7767

547.2765

558.7750

TD

481.9259

490.0950

498.7635

507.7332

516.8405

525.9066

6.0

TID

503.5508

515.9382

529.1391

542.8510

556.8042

570.6840

TD

481.9711

491.9193

502.4538

513.3313

524.3387

535.2285

4.0

TID

503.7081

519.5416

536.3883

553.8568

571.5649

589.0330

TD

482.0982

494.7982

508.2006

521.9910

535.8672

549.4501

8.0

TABLE 3.16 Comparisons of Buckling Temperature DT (K) for (FGM=Ti-6Al-4V=FGM) Sandwich Plates under Uniform Temperature Rise (b=h ¼ 20, a=b ¼ 1.0, T0 ¼ 300 K) hH=hF

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

TID

654.3356

650.9921

644.7942

637.1305

628.6901

619.7468

TD

621.3503

591.1686

567.3193

548.8891

533.7027

521.1356

6.0

TID

654.3357

649.6246

641.5507

631.8915

621.4725

610.5759

TD

621.3504

585.1382

558.8517

538.8232

523.0273

510.2279

4.0

TID TD

654.3357 621.3504

646.9642 576.1403

635.6199 546.6052

622.7022 525.0375

609.1954 508.7258

595.3411 495.3070

8.0

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temperature of (FGM=Ti-6Al-4V=FGM) sandwich plate is larger than that of (FGM=SUS304=FGM) one. It is found that the increase is about þ14% for the (FGM=SUS304=FGM) sandwich plate, and about 20% for the (FGM=Ti- 6Al4V=FGM) one with hH=hF ¼ 4, from N ¼ 0 to N ¼ 5, under TD case. Figure 3.15 shows the effect of material properties on the postbuckling behavior of the (FGM=SUS304=FGM) sandwich plate with hH=hF ¼ 4 5000 (FGM/SUS304/FGM) T0 = 300 K, N = 2.0, hH/hF = 4.0 b = 1.0, b/h = 20, (m, n) = (1, 1) ∆TU = ∆TL = 200 K

Px (kN)

4000 3000

I II

I: TID II: TD

2000 1000

W ∗/h = 0.0 W ∗/h = 0.1

0

2

0

4

(a)

6

8

10

12

W (mm) 5000

Px (kN)

4000

3000

(FGM/SUS304/FGM) T0 = 300 K, N = 2.0, hH/hF = 4.0 b = 1.0, b/h = 20, (m, n) = (1, 1) ∆TU = ∆TL= 200 K

I

2000

W ∗/h = 0.0 W ∗/h = 0.1

1000

0 -1 (b)

II

I: TID II: TID

0

1

2 ∆x (mm)

3

4

5

FIGURE 3.15 Effects of material properties on the postbuckling behavior of (FGM=SUS304=FGM) sandwich plates subjected to uniaxial compression under uniform temperature ﬁeld: (a) load deﬂection; (b) load shortening.

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Postbuckling of Shear Deformable FGM Plates

93

and N ¼ 2.0 subjected to uniaxial compression under uniform temperature ﬁeld DTU ¼ DTL ¼ 200 K, and under two cases of thermoelastic material properties, i.e., TID and TD. It can be seen that the postbuckling equilibrium path becomes lower when the TD properties are taken into account. Figure 3.16 shows temperature changes on the postbuckling behavior of the same plate subjected to uniaxial compression under heat conduction 5000

4000

(FGM/SUS304/FGM) T0 = 300 K, N = 2.0, hH/hF = 4.0 β = 1.0, b/h = 20, (m, n) = (1, 1) 1: ∆TU = ∆TL= 100 K

Px (kN)

3000

2000

1000

0 0

2: ∆TU = 200 K, ∆TL= 100 K 3: ∆TU = 300 K, ∆TL= 100 K 1 2 3

W ∗/h = 0.0 W ∗/h = 0.1

2

4

(a)

5000

4000

6 W (mm)

8

10

12

(FGM/SUS304/FGM) T0 =300 K, N = 2.0, hH/hF = 4.0 β = 1.0, b/h = 20, (m, n) = (1, 1)

Px (kN)

1: ∆TU = ∆TL = 100 K 3000

2: ∆TU = 200 K, ∆TL = 100 K 3: ∆TU = 300 K, ∆TL = 100 K

2000

1000

0 −1 (b)

32 0

W ∗/h = 0.0 W ∗/h = 0.1

1 1

2 ∆x (mm)

3

4

5

FIGURE 3.16 Thermal effects on the postbuckling behavior of (FGM=SUS304=FGM) sandwich plates subjected to uniaxial compression: (a) load deﬂection; (b) load shortening.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

and TD case. It can be seen that the postbuckling strength is decreased with increase in temperature. It can be seen that the initial deﬂection is not zerovalued and an initial extension occurs (curves 2 and 3 in the ﬁgure) when the heat conduction is put into consideration. The results conﬁrm that no buckling loads exist and the plate will buckle at the onset of edge compression when under heat conduction, and the shape of the load–deﬂection curves for perfect plates appears similar to that for plates with an initial geometric imperfection. Note that DTU ¼ DTL ¼ 100 K means uniform temperature ﬁeld, so that the buckling load still exists (curve 1 in the ﬁgure). Figure 3.17 shows the effect of the volume fraction index N ( ¼ 0.2, 2.0, and 5.0) on the postbuckling behavior of the (FGM=SUS304=FGM) sandwich plate with hH=hF ¼ 4 subjected to uniaxial compression under heat conduction DTU ¼ 200 K, DTL ¼ 100 K and TD case. It can be seen that the initial deﬂection is decreased, but the initial extension is almost the same when the volume fraction index N is increased. It is found that the increase of the volume fraction index N yields an increase of postbuckling strength. Figures 3.18 and 3.19 are thermal postbuckling results for the (FGM= SUS304=FGM) sandwich plate analogous to the compressive postbuckling results of Figures 3.15 and 3.17, which are for the thermal loading case of heat conduction with TL ¼ 300 K. The results conﬁrm that for the case of heat conduction, the thermal postbuckling path is no longer of the bifurcation type.

5000

(FGM/SUS304/FGM) T0 = 300 K, hH/hF = 4.0

4000

β = 1.0, b/h= 20, (m, n) = (1, 1)

1

∆TU = 200 K, ∆TL =100 K

Px (kN)

2

1: N = 5.0

3000

3

2: N = 2.0 3: N = 0.2 2000

1000

W ∗/h = 0.0 W ∗/h = 0.1

0 (a)

0

2

4

6

8

10

12

W (mm)

FIGURE 3.17 Effects of volume fraction index N on the postbuckling behavior of (FGM=SUS304=FGM) sandwich plates subjected to uniaxial compression under heat conduction: (a) load deﬂection;

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Postbuckling of Shear Deformable FGM Plates 5000

4000

95

(FGM/SUS304/FGM) T0 = 300 K, hH/hF = 4.0 β = 1.0, b/h= 20, (m, n) = (1, 1)

1 2 3

∆TU = 200 K, ∆TL =100 K

Px (kN)

3000

1: N = 5.0 2: N = 2.0 3: N = 0.2

2000 W ∗/h = 0.0

1000

W ∗/h = 0.1

0 –1

0

1

(b)

2 ∆x (mm)

3

4

5

FIGURE 3.17 (continued) (b) load shortening.

2000 Heat conduction (FGM/SUS304/FGM) TL = 300 K, N = 2.0, hH/hF = 4.0

1500

∆T (K)

β = 1.0, b/h = 20, (m, n) = (1, 1)

TID TD

1000

500

W ∗/h = 0.0 W ∗/h = 0.05

0

0

2

4 W (mm)

6

8

FIGURE 3.18 Effects of material properties on the thermal postbuckling behavior of (FGM=SUS304=FGM) sandwich plates under heat conduction.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 2000 Heat conduction (FGM/SUS304/FGM) TL = 300 K, hH/hF = 4.0 β = 1.0, b/h= 20, (m, n) = (1, 1)

∆T (K)

1500

1: N = 0.2 2: N = 2.0 3: N = 5.0

1000

3

500

0

2 1

W ∗/h = 0.0 W ∗/h = 0.05 0

2

4

6

8

W (mm)

FIGURE 3.19 Effects of volume fraction index N on the thermal postbuckling behavior of (FGM=SUS304=FGM) sandwich plates under heat conduction.

References Bhimaraddi A. and Chandrashekhara K. (1993), Nonlinear vibrations of heated antisymmetric angle-ply laminated plates, International Journal of Solids and Structures, 30, 1255–1268. Birman V. (1995), Stability of functionally graded hybrid composite plates, Composites Engineering, 5, 913–921. Chen X.L. and Liew K.M. (2004), Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads, Smart Materials and Structures, 13, 1430–1437. Chen X.L., Zhao Z.Y., and Liew K.M. (2008), Stability of piezoelectric FGM rectangular plates subjected to non-uniformly distributed load, heat and voltage, Advances in Engineering Software, 39, 121–131. Dym C.L. (1974), Stability Theory and its Applications to Structural Mechanics, Noordhoff, Leyden. Feldman E. and Aboudi J. (1997), Buckling analysis of functionally graded plates subjected to uniaxial loading, Composite Structures, 38, 29–36. Ganapathi M. and Prakash T. (2006), Thermal buckling of simply supported functionally graded skew plates, Composite Structures, 74, 247–250. Ganapathi M., Prakash T., and Sundararajan N. (2006), Inﬂuence of functionally graded material on buckling of skew plates under mechanical loads, Journal of Engineering Mechanics ASCE, 132, 902–905. Javaheri R. and Eslami M.R. (2002a), Buckling of functionally graded plates under in-plane compressive loading, ZAMM, 82, 277–283.

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Javaheri R. and Eslami M.R. (2002b), Thermal buckling of functionally graded plates, AIAA Journal, 40, 162–169. Javaheri R. and Eslami M.R. (2002c), Thermal buckling of functionally graded plates based on higher order theory, Journal of Thermal Stresses, 25, 603–625. Kamiya N. and Fukui A. (1982), Finite deﬂection and postbuckling behavior of heated rectangular plates with temperature-dependent properties, Nuclear Engineering and Design, 72, 415–420. Leissa A.W. (1986), Conditions for laminated plates to remain ﬂat under inplane loading, Composite Structures, 6, 261–270. Li S.-R., Zhang J.-H., and Zhao Y.-G. (2007), Nonlinear thermomechanical postbuckling of circular FGM plate with geometric imperfection, Thin-Walled Structures, 45, 528–536. Liew K.M., Yang J., and Kitipornchai S. (2003), Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading, International Journal of Solids and Structures, 40, 3869–3892. Liew K.M., Yang J., and Kitipornchai S. (2004), Thermal post-buckling of laminated plates comprising functionally graded materials with temperature-dependent properties, Journal of Applied Mechanics ASME, 71, 839–850. Ma L.S. and Wang T.J. (2003a), Axisymmetric post-buckling of a functionally graded circular plate subjected to uniformly distributed radial compression, Materials Science Forum, 423–425, 719–724. Ma L.S. and Wang T.J. (2003b), Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings, International Journal of Solids and Structures, 40, 3311–3330. Ma L.S. and Wang T.J. (2004), Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory, International Journal of Solids and Structures, 41, 85–101. Na K.-S. and Kim J.-H. (2004), Three-dimensional thermal buckling analysis of functionally graded materials, Composites Part B, 35, 429–437. Na K.-S. and Kim J.-H. (2006a), Thermal postbuckling investigations of functionally graded plates using 3-D ﬁnite element method, Finite Elements in Analysis and Design, 42, 749–756. Na K.-S. and Kim J.-H. (2006b), Three-dimensional thermomechanical buckling analysis of functionally graded composite plates, Composite Structures, 73, 413–422. Naei M.H., Masoumi A., and Shamekhi A. (2007), Buckling analysis of circular functionally graded material plate having variable thickness under uniform compression by ﬁnite-element method, Proceedings of the Institution of Mechanical Engineers Part C—Journal of Mechanical Engineering Science, 221, 1241–1247. Najaﬁzadeh M.M. and Eslami M.R. (2002a), Buckling analysis of circular plates of functionally graded materials under uniform radial compression, International Journal of Mechanical Sciences, 44, 2479–2493. Najaﬁzadeh M.M. and Eslami M.R. (2002b), First-order-theory-based thermoelastic stability of functionally graded material circular plates, AIAA Journal, 40, 1444–1450. Najaﬁzadeh M.M. and Heydari H.R. (2004a), Reﬁned theory for thermoelastic stability of functionally graded circular plates, Journal of Thermal Stresses, 27, 857–880. Najaﬁzadeh M.M. and Heydari H.R. (2004b), Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory, European Journal of Mechanics A=Solids, 23, 1085–1100.

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Najaﬁzadeh M.M. and Heydari H.R. (2008), An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression, International Journal of Mechanical Sciences, 50, 603–612. Oh I.K., Han J.H., and Lee I. (2000), Postbuckling and vibration characteristics of piezolaminated composite plate subjected to thermo-piezoelectric loads, Journal of Sound and Vibration, 233, 19–40. Park J.S. and Kim J.-H. (2006), Thermal postbuckling and vibration analyses of functionally graded plates, Journal of Sound and Vibration, 289, 77–93. Prakash T., Singha M.K., and Ganapathi M. (2008), Thermal postbuckling analysis of FGM skew plates, Engineering Structures, 30, 22–32. Qatu M.S. and Leissa A.W. (1993), Buckling or transverse deﬂections of unsymmetrically laminated plates subjected to in-plane loads, AIAA Journal, 31, 189–194. Shariat B.A.S. and Eslami M.R. (2005), Effect of initial imperfections on thermal buckling of functionally graded plates, Journal of Thermal Stresses, 28, 1183–1198. Shariat B.A.S. and Eslami M.R. (2006), Thermal buckling of imperfect functionally graded plates, International Journal of Solids and Structures, 43, 4082–4096. Shen H.-S. (2002), Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments, International Journal of Mechanical Sciences, 44, 561–584. Shen H.-S. (2005), Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings, International Journal of Solids and Structures, 42, 6101–6121. Shen H.-S. (2007a), Nonlinear thermal bending response of FGM plates due to heat conduction, Composites Part B, 38, 201–215. Shen H.-S. (2007b), Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties, International Journal of Mechanical Sciences, 49, 466–478. Shen H.-S. and Li S.-R. (2008), Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties, Composites Part B, 39, 332–344. Shukla K.K., Kumar K.V.R., Pandey R., and Nath Y. (2007), Postbuckling response of functionally graded rectangular plates subjected to thermo-mechanical loading, International Journal of Structural Stability and Dynamics, 7, 519–541. Woo J., Meguid S.A., Stranart J.C., and Liew K.M. (2005), Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells, International Journal of Mechanical Sciences, 47, 1147–1171. Wu L. (2004), Thermal buckling of simply supported moderately thick rectangular FGM plate, Composite Structures, 64, 211–218. Wu T.-L., Shukla K.K., and Huang J.H. (2007), Post-buckling analysis of functionally graded rectangular plates, Composite Structures, 81, 1–10. Yamaki N. (1961), Experiments on the postbuckling behavior of square plates loaded in edge compression, Journal of Applied Mechanics ASME, 28, 238–244. Yang J., Liew K.M., and Kitipornchai S. (2006), Imperfection sensitivity of the postbuckling behavior of higher-order shear deformable functionally graded plates, International Journal of Solids and Structures, 43, 5247–5266. Yang J. and Shen H.-S. (2003), Nonlinear analysis of functionally graded plates under transverse and in-plane loads, International Journal of Non-Linear Mechanics, 38, 467–482.

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4 Nonlinear Vibration of Shear Deformable FGM Plates

4.1

Introduction

The external loads applied to a plate invariably change with time. If the variations in time are small and occur over an extended interval, the inertial effects may be neglected and the behavior of the plate can be approximately determined from considerations of equilibrium and material properties. However, in some modern engineering constructions, like aircraft, missiles, and launch vehicles, rapid time variations of loadings occur that must be considered in formulating structural design. In such cases, inertial effects must be taken into account and the dynamic behavior of the plate must be treated as a function of time. Many studies have been reported on the free and forced vibration of FGM plates, for example, Yang and Shen (2001, 2002), Cheng and Reddy (2003), Vel and Batra (2004), Kim (2005), Elishakoff et al. (2005), Prakash and Ganapathi (2006), Efraim and Eisenberger (2007), and Li et al. (2008). According to mixture rules, Abrate (2006) found that the natural frequencies of FGM plates are proportional to those of the corresponding homogeneous plate and concluded that only one case is needed to determine the proportionality constant, and direct analysis is unnecessary. In most conditions of severe environments, when the plate deﬂection-tothickness ratio is greater than 0.4, the nonlinearity is very important and the nonlinear dynamic equations of plates are required to perform the analysis. Praveen and Reddy (1998) analyzed the nonlinear static and dynamic response of functionally graded ceramic–metal plates in a steady temperature ﬁeld and subjected to dynamic transverse loads by ﬁnite element method (FEM). Reddy (2000) developed both theoretical and ﬁnite element formulations for thick FGM plates according to the HSDPT, and studied the nonlinear dynamic response of FGM plates subjected to suddenly applied uniform pressure. Yang et al. (2003) presented a large amplitude vibration analysis of an initially stressed FGM plate with surface-bonded piezoelectric layers by using a semianalytical method based on 1D differential quadrature and Galerkin technique. Chen (2005), Chen et al. (2006), Fung and Chen (2006), and Chen and Tan (2007) studied the large amplitude vibration of 99

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an initially stressed FGM plate with or without initial geometric imperfections. In their studies, the initial stress was taken to be a combination of pure bending stress and an extensional stress in the plane of the plate, and the formulations were based on the FSDPT and classical plate theory (CPT), respectively. Also, Allahverdizadeh et al. (2008a–c) studied the nonlinear free and forced vibration of FGM circular thin plates based on the CPT. However, in the references cited above (Praveen and Reddy 1998, Reddy 2000, Yang et al. 2003, Chen 2005, Chen et al. 2006, Chen and Tan 2007, Allahverdizadeh et al. 2008a–c), the material properties were either assumed to be independent of temperature or considered in a constant temperature environment (T ¼ 300 K). Since FGMs always serve in the high-temperature environments, the materials properties of FGM plates must be temperatureand position-dependent. Kitipornchai et al. (2004) and Yang and Huang (2007) studied nonlinear free and forced vibration of imperfect FGM laminated plates with various boundary conditions and with temperature-dependent material properties, respectively. On the other hand, ceramics and the metals used in FGM do store different amounts of heat, and therefore the heat conduction usually occurs. This leads to a nonuniform distribution of temperature through the plate thickness, but it is not accounted for in the above study. Also recently, Huang and Shen (2004, 2006) provided nonlinear free and forced vibration analysis of shear deformable FGM plates without or with surface-bonded piezoelectric layers in thermal environments. Xia and Shen (2008) provided small- and large-amplitude vibration analysis of compressively and thermally postbuckled sandwich plates with FGM face sheets in thermal environments. In these studies, heat conduction and temperaturedependent material properties were both taken into account. Sundararajan et al. (2005) calculated frequencies for nonlinear free ﬂexural vibration of functionally graded rectangular and skew plates under thermal environments. In their studies, the material properties were based on the Mori– Tanaka scheme, and a remarkable synergism between the Mori–Tanaka scheme and the rule of mixture was found.

4.2

Nonlinear Vibration of FGM Plates in Thermal Environments

Here, we consider an FGM plate of length a, width b, and thickness h, which is made from a mixture of ceramics and metals. We assume that the composition is varied from the top to the bottom surface, i.e., the top surface (Z ¼ h=2) of the plate is ceramic-rich, whereas the bottom surface (Z ¼ h=2) is metal-rich. Note that in Sections 4.2 and 4.3 the Z is in the direction of the upward normal to the middle surface. A simple power law exponent of the volume fraction distribution is used to provide a measure of the amount

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Nonlinear Vibration of Shear Deformable FGM Plates

101

of ceramic and metal in the FGM. In the present case, the volume fraction of ceramic, instead of Vm, is deﬁned by Vc (Z) ¼

2Z þ h N 2h

(4:1)

It is assumed that the effective Young’s modulus Ef and thermal expansion coefﬁcient af of the FGM plate are temperature dependent, whereas the mass density rf and thermal conductivity kf are independent of the temperature. Poisson’s ratio nf depends weakly on temperature change and is assumed to be a constant. As long as the volume fraction is known, the effective material properties can be obtained easily using the rule of mixture

2Z þ h N þ Em (T) 2h 2Z þ h N þ am (T) af (Z, T) ¼ [ac (T) am (T)] 2h 2Z þ h N þ rc rf (Z) ¼ (rc rm ) 2h 2Z þ h N þ km k(Z) ¼ (kc km ) 2h Ef (Z, T) ¼ [Ec (T) Em (T)]

(4:2a) (4:2b) (4:2c) (4:2d)

We assume that the temperature variation occurs in the thickness direction only and 1D temperature ﬁeld is assumed to be constant in the XY plane of the plate. In such a case, the temperature distribution along the thickness can be obtained by solving a steady-state heat transfer equation: d dT k(Z) ¼0 dZ dZ

(4:3)

This equation is solved by imposing boundary condition of T ¼ TU at Z ¼ h=2 and T ¼ TL at Z ¼ h=2. The solution of this equation, by means of polynomial series, is T(Z) ¼ TL þ (TU TL )h(Z)

(4:4)

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

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1 h(Z) ¼ C

" 2Z þ h kcm 2Z þ h Nþ1 k2cm 2Z þ h 2Nþ1 þ (N þ 1)km (2N þ 1)k2m 2h 2h 2h k3cm 2Z þ h 3Nþ1 k4cm 2Z þ h 4Nþ1 þ (3N þ 1)k3m (4N þ 1)k4m 2h 2h # 5Nþ1 k5cm 2Z þ h (4:5a) (5N þ 1)k5m 2h C¼1 þ

kcm k2cm k3cm þ 2 (N þ 1)km (2N þ 1)km (3N þ 1)k3m

k4cm k5cm 4 (4N þ 1)km (5N þ 1)k5m

(4:5b)

where kcm ¼ kc km. The plate is assumed to be geometrically perfect, and is subjected to a transverse dynamic load q(X, Y, t) in thermal environments. Hence, the general von Kármán-type equations can be expressed by Equations 1.33 through 1.36, and the forces, moments, and higher order moments caused by elevated temperature are deﬁned by Equation 1.28. It is assumed that all four edges are simply supported with no in-plane displacements. In such a case, the boundary conditions can be expressed by Equation 2.6 for immovable edges. Introducing dimensionless quantities Equation 2.8, and sﬃﬃﬃﬃﬃ pt E0 I1 E0 a2 4E0 (I5 I1 I4 I2 ) , g170 ¼ 2 , g171 ¼ , t¼ * * r0 p r0 D11 a 3r0 h2 I1 D11 (g 80 , g 90 , g10 ) ¼ (I 8 , I 5 , I 3 )

E0 * r0 D11

(4:6)

where E0 and r0 are the reference values of Em and rm, respectively, at room temperature (T0 ¼ 300 K) and ATx (¼ATy ), DTx (¼DTy ), and FTx (¼FTy ) are redeﬁned by "

ATx

DTx

FTx

ATy

DTy

FTy

#

h=2 ð

T1 ¼ h=2

Ax Ay

(1, Z, Z3 )DT(Z)dZ

(4:7)

where T1 ¼ (TU þ TL 2T0)=2. The nonlinear motion equations can then be written in dimensionless form as L11 (W) L12 (Cx ) L13 (Cy ) þ g 14 L14 (F) L16 (MT ) ! €y €x @ C @ C 2 € þg þb ¼ g14 b L(W, F) þ L17 (W) þ lq 80 @x @y

(4:8)

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1 L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) g 24 L24 (W) ¼ g 24 b2 L(W, W) (4:9) 2 € @W € x (4:10) þ g10 C L31 (W) þ L32 (Cx ) L33 (Cy ) þ g14 L34 (F) L36 (ST ) ¼ g90 @x € @W € y (4:11) þ g 10 C L41 (W) L42 (Cx ) þ L43 (Cy ) þ g14 L44 (F) L46 (ST ) ¼ g90 b @y where L17 () ¼ g 170 þ g 171

2 @2 2 @ þ g b 171 @x2 @y2

(4:12)

and all other nondimensional linear operators Lij() and nonlinear operator L() are deﬁned by Equation 2.14. The boundary conditions can be written in dimensionless form as x ¼ 0, p: W ¼ Cy ¼ 0

(4:13a)

ðp ðp

2 @Cy @2F @2F @Cx @2W 2@ W g þ g g b g þ g b þ g g 5 24 511 233 24 611 244 @x @y @y2 @x2 @x2 @y2 0 # 1 @W 2 þ (g 224 gT1 g5 gT2 )T1 dx dy ¼ 0 (4:13b) g24 2 @x g224 b2

0

y ¼ 0, p: W ¼ Cx ¼ 0 ðp ðp 0 0

(4:13c)

2 2 @Cy @2F @Cx @2W 2@ F 2@ W g b þ g g b g þ g b þ g g 5 24 220 522 24 240 622 @x @y @x2 @y2 @x2 @y2

1 @W 2 g24 b2 þ (gT2 g 5 gT1 )T1 dy dx ¼ 0 2 @y

(4:13d)

We assume that the solutions of Equations 4.8 through 4.11 can be expressed as ~ x (x, y, t) W(x, y, t) ¼ W*(x, y) þ W ~ x (x, y, t) Cx (x, y, t) ¼ Cx* (x, y) þ C ~ y (x, y, t) Cy (x, y, t) ¼ C*y (x, y) þ C ~ y, t) F(x, y, t) ¼ F*(x, y) þ F(x,

(4:14)

where W*(x, y) is an initial deﬂection due to initial thermal bending moment ~ W(x, y, t) is an additional deﬂection

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Cx*(x, y), Cy*(x, y), and F*(x, y) are the midplane rotations and stress function ~ y(x, y, t), and F(x, ~ x(x, y, t), C ~ y, t) are deﬁned corresponding to W*(x, y). C ~ analogously to Cx*(x, y), Cy*(x, y), and F*(x, y), but is for W(x, y, t). Due to the bending–stretching coupling effect in the FGM plate, the thermal preload will bring about deﬂections and bending curvatures which have signiﬁcant inﬂuences on the plate vibration characteristics. To account for this effect, the previbration solutions W*(x, y), Cx*(x, y), Cy*(x, y), and F*(x, y) are sought at the ﬁrst step from the following nonlinear equations: L11 (W*) L12 (C*x ) L13 (C*y ) þ g14 L14 (F*) L16 (MT ) @ 2 W* @ 2 W* 2 ¼ g 14 b2 L(W*, F*) þ py þ g 14 b px @x2 @y2

(4:15)

1 L21 (F*) þ g24 L22 (Cx* ) þ g 24 L23 (Cy* ) g24 L24 (W*) ¼ g24 b2 L(W*, W*) 2 (4:16) L31 (W*) þ L32 (C*x ) L33 (C*y ) þ g 14 L34 (F*) L36 (ST ) ¼ 0

(4:17)

L41 (W*) L42 (C*x ) þ L43 (C*y ) þ g 14 L44 (F*) L46 (ST ) ¼ 0

(4:18)

In Equation 4.15, px and py are edge compressive stresses induced by temperature rise with edge restraints. The solutions of Equations 4.15 through 4.18 can be assumed to be as X

W*(x, y) ¼

X

wkl sin kx sin ly

k¼1,3,... l¼1,3,...

X

C*x (x, y) ¼

X

(cx )kl cos kx sin ly

k¼1,3,... l¼1,3,...

X

C*y (x, y) ¼

X

(cy )kl sin kx cos ly

(4:19)

k¼1,3,... l¼1,3,...

F*(x, y) ¼

X X 1 (0) 2 2 B00 y þ b(0) x fkl sin kx sin ly þ 00 2 k¼1,3,... l¼1,3,...

We then expand the constant thermal bending moments in the double Fourier sine series as "

MTx

STx

MTy

STy

#

" ¼

M(0) x

S(0) x

M(0) y

S(0) y

#

X k¼1,3,...

X 1 sin kx sin ly kl l¼1,3,...

(4:20)

Substituting Equations 4.19 and 4.20 into Equations 4.15 through 4.18 and applying Galerkin procedure to Equations 4.15 and 4.16, Wkl, (cx)kl, (cy)kl, and fkl can easily be determined. The detailed expressions are given in Appendix I.

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Nonlinear Vibration of Shear Deformable FGM Plates

105

~ y, t), Then an initially stressed FGM plate is under consideration and W(x, ~ ~ ~ Cx(x, y, t), Cy(x, y, t), and F(x, y, t) satisfy the nonlinear equations: ~ x ) L13 (C ~ y ) þ g L14 (F) ~ L12 (C ~ þ W*, F) ~ F*)] ~ ¼ g14 b2 [L(W ~ þ L(W, L11 (W) 14 ! € € ~y ~x @C @C € ~ þg þb þ lq (4:21) þ L17 W 80 @x @y ~ x ) þ g L23 (C ~ y ) g L24 (W) ~ ¼ 1 g24 b2 L(W ~ þ 2W*, W) ~ (4:22) ~ þ g 24 L22 (C L21 (F) 24 24 2

€~ @W €~ þ g 10 C x @x €~ €~ ~ x ) þ L43 (C ~ y ) þ g L44 (F) ~ L42 (C ~ ¼ g 90 b @ W þ g 10 C L41 (W) y 14 @y ~ x ) L33 (C ~ y ) þ g L34 (F) ~ þ L32 (C ~ ¼ g 90 L31 (W) 14

(4:23) (4:24)

The initial conditions are assumed to be ~ @W ¼ ¼0 t¼0 @t t¼0 ~ ~ x ¼ @ Cx ¼ 0 C t¼0 @t t¼0 ~ ~ y ¼ @ Cy ¼ 0 C t¼0 @t ~ W

(4:25a)

(4:25b)

(4:25c)

t¼0

A perturbation technique is now used to solve Equations 4.21 through 4.24. The essence of this procedure, in the present case, is to assume that X ~ y, t~, «) ¼ «j Wj (x, y, t~) W(x, j¼1

~ y, t~, «) ¼ F(x,

X

«i Fj (x, y, t~)

j¼1

~ x (x, y, t~, «) ¼ C

X

«j Cxj (x, y, t~)

j¼1

~ y (x, y, t~, «) ¼ C

X

(4:26)

«j Cyj (x, y, t~)

j¼1

lq (x, y, t~, «) ¼

X

«j lj (x, y, t~)

j¼1

where « is a small perturbation parameter. Here, we introduce an important parameter t~ ¼ «t to improve perturbation procedure for solving nonlinear dynamic problem.

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Substituting Equation 4.26 into Equations 4.21 through 4.24 and collecting the terms of the same order of «, we obtain a set of perturbation equations which can be written, for example, as O(«0): L14 ( f0 ) ¼ 0

(4:27a)

L21 ( f0 ) ¼ 0

(4:27b)

L34 ( f0 ) ¼ 0

(4:27c)

L44 ( f0 ) ¼ 0

(4:27d)

O(«1): L11 (w1 ) L12 (cx1 ) L13 (cy1 ) þ g 14 L14 ( f1 ) ¼ g14 b2 [L(w1 þ W*, f0 ) þ L(w1 , F*)] þ l1

(4:28a)

L21 ( f1 ) þ g 24 L22 (cx1 ) þ g24 L23 (cy1 ) g 24 L24 (w1 ) ¼ 0

(4:28b)

L31 (w1 ) þ L32 (cx1 ) L33 (cy1 ) þ g14 L34 ( f1 ) ¼ 0

(4:28c)

L41 (w1 ) L42 (cx1 ) þ L43 (cy1 ) þ g14 L44 ( f1 ) ¼ 0

(4:28d)

O(«2): L11 (w2 ) L12 (cx2 ) L13 (cy2 ) þ g 14 L14 ( f2 ) ¼ g14 b2 [L(w2 , f0 þ F*) þ L(w1 þ W*, f1 )] þ l2

(4:29a)

L21 ( f2 ) þ g 24 L22 (cx2 ) þ g24 L23 (cy2 ) g 24 L24 (w2 ) 1 ¼ g24 b2 L(w1 þ 2W*, w1 ) 2

(4:29b)

L31 (w2 ) þ L32 (cx2 ) L33 (cy2 ) þ g14 L34 ( f2 ) ¼ 0

(4:29c)

L41 (w2 ) L42 (cx2 ) þ L43 (cy2 ) þ g14 L44 ( f2 ) ¼ 0

(4:29d)

O(«3): L11 (w3 ) L12 (cx3 ) L13 (cy3 ) þ g14 L14 ( f3 ) ¼ g 14 b2 [Lðw3 , f0 þ F*Þ þ L(w2 , f1 ) þ Lðw1 þ W*, f2 Þ] þ l3

(4:30a)

L21 ( f3 ) þ g 24 L22 (cx3 ) þ g24 L23 (cy3 ) g 24 L24 (w3 ) 1 ¼ g24 b2 Lðw1 þ 2W*, w2 Þ 2

(4:30b)

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Nonlinear Vibration of Shear Deformable FGM Plates

107

€1 @w þ g10 c€x1 @x €1 @w þ g10 c€y1 L41 (w3 ) L42 (cx3 ) þ L43 (cy3 ) þ g 14 L44 ( f3 ) ¼ g90 b @y L31 (w3 ) þ L32 (cx3 ) L33 (cy3 ) þ g 14 L34 ( f3 ) ¼ g 90

(4:30c) (4:30d)

To solve these perturbation equations of each order, the amplitudes of the terms wj(x, y), fj(x, y), cxj(x, y), and cyj(x, y) can be determined step by step. As a result, we obtain asymptotic solutions, up to third order, as ~ y, t) ¼ «[w1 (t) þ g1 w € 1 (t)] sin mx sin ny þ («w1 (t))3 [ag311 sin mx sin ny W(x, (4:31) þ g331 sin 3mx sin ny þ g313 sin mx sin 3ny] þ O(«4 ) h i ~ x (x, y, t) ¼ « g(1,1) w1 (t) þ g2 w € 1 (t) cos mx sin ny þ g12 («w1 (t))2 sin 2mx C 11 h (3,1) þ («w1 (t))3 ag(1,1) 11 g311 cos mx sin ny þ g11 g331 cos 3mx sin ny i þ g(1,3) g cos mx sin 3ny þ O(«4 ) (4:32) 313 11 h i ~ y (x, y, t) ¼ « g(1,1) w1 (t) þ g3 w € 1 (t) sin mx cos ny þ g22 («w1 (t))2 sin 2ny C 21 h (3,1) þ («w1 (t))3 ag(1,1) 21 g311 sin mx cos ny þ g21 g331 sin 3mx cos ny i 4 þ g(1,3) (4:33) 21 g313 sin mx cos 3ny þ O(« ) h i ~ y, t) ¼ « g(1,1) w1 (t) þ g4 w € F(x, (t) sin mx sin ny («w1 (t))2 1 31 (2) 2 2 B(2) y =2 þ b x =2 g cos 2ny g cos 2mx 402 420 00 00 h (1,1) (3,1) þ («w1 (t))3 ag31 g311 sin mx sin ny þ g31 g331 sin 3mx sin ny i þ g(1,3) g sin mx sin 3ny þ O(«4 ) (4:34) 313 31 € 1 (t)] sin mx sin ny lq (x, y, t) ¼ «[g41 w1 (t) þ g43 w þ («w1 (t))2 (g441 cos 2mx þ g442 cos 2ny) g14 b2 («w1 (t))2

XX k

wkl

l

(2) 2 2 2 2 2 2 B(2) k þ b l 4k n g cos 2ny 4l m g cos 2mx sin kx sin ly 402 420 00 00 þ ag42 («w1 (t))3 sin mx sin ny þ O(«4 )

(4:35)

Note that in Equations 4.31 through 4.35 t~ is replaced by t, and for the case of free vibration a ¼ 0, a ¼ 1, otherwise a ¼ 1, a ¼ 0. Coefﬁcients (i,j) (i,j) (i,j) g11 , g21 , g31 (i, j ¼ 1, 3), etc. are given in detail in Appendix J. Multiplying Equation 4.34 by (sin mx sin ny) and integrating over the entire plate area, one has

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108

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

g43

d2 («w1 ) þ g41 («w1 ) þ g44 («w1 )2 þ ag42 («w1 )3 ¼ lq (t) dt 2

(4:36)

where 4 lq (t) ¼ 2 p

4.2.1

ðp ðp lq (x, y, t) sin mx sin ny dx dy

(4:37)

0 0

Free Vibration

When a ¼ 1, lq (t) ¼ 0, Equation 4.36 becomes the free vibration equation of the plate. The nonlinear frequency of the plates can be expressed as 1=2 9g42 g41 10g244 2 A vNL ¼ vL 1 þ 12g241

(4:38)

where A ¼ W max =h is the amplitude to thickness ratio, and vL ¼ [g41 =g43 ]1=2 is the dimensionless linear frequency, from which the corresponding linear frequency can be expressed as vL ¼ vL (p=a)(E0 =r0 )1=2 , where E0 and r0 are deﬁned as in Equation 4.6. 4.2.2

Forced Vibration

When the forced vibration is under consideration, we take a ¼ 0. In such a case, Equation 4.36 can be rewritten as € 1 (t) þ «w1 (t)v2L þ «w

lq (t) g44 («w1 (t))2 þ O «4 ¼ g43 g43

(4:39)

If zero-valued initial conditions prevail, i.e., w1(0) ¼ w_ 1(0) ¼ 0, Equation 4.39 may then be solved by using the Runge–Kutta iteration scheme (Pearson 1986): («w1 )iþ1 ¼ («w1 )i þ Dt(«w_ 1 )i þ («w_ 1 )iþ1

(Dt)2 (L1 þ L2 þ L3 ) 6

Dt (L1 þ 2L2 þ 2L3 þ L4 ) ¼ («w_ 1 )i þ 6

where Dt is the time step, and

(4:40)

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Nonlinear Vibration of Shear Deformable FGM Plates

109

L1 ¼ f (t i , («w1 )) Dt Dt(«w_ 1 )i , («w1 )i þ L2 ¼ f t i þ 2 2

! Dt Dt(«w_ 1 )i (Dt)2 , («w1 )i þ þ L1 L3 ¼ f t i þ 2 2 4 ! (Dt)2 L2 L4 ¼ f t i þ Dt, («w1 )i þ Dt(«w_ 1 )i þ 2

(4:41)

where f (t, x) ¼ v2L x

g44 2 lq (t) x þ g43 g43

(4:42)

As a result, the solution of Equation 4.39 is obtained numerically. Resubstituting it into Equations 4.31 through 4.35, both displacement and stress function are determined. Next, substituting Equation 4.14 into boundary (0) (2) (2) conditions (Equation 4.13), the coefﬁcients B(0) 00 , b00 , B00 , and b00 are then determined as given in Appendix J. For numerical illustrations, two sets of material mixture are considered. One is zirconium oxide and titanium alloy, referred to as ZrO2=Ti-6Al-4V, and the other is silicon nitride and stainless steel, referred to as Si3N4=SUS304. The upper surface of these two FGM plates is ceramic-rich and the lower surface is metal-rich. The thickness and side of the square plate are h ¼ 0.025 m and a ¼ 0.2 m, respectively. The mass density and thermal conductivity are r ¼ 3000 kg m3, k ¼ 1.80 W mK1 for ZrO2; r ¼ 4429 kg m3, k ¼ 7.82 W mK1 for Ti-6Al-4V; r ¼ 2370 kg m3, k ¼ 9.19 W mK1 for Si3N4; and r ¼ 8166 kg m3, k ¼ 12.04 W mK1 for SUS304. Young’s modulus and thermal expansion coefﬁcient of these materials are assumed to be nonlinear function of temperature of Equation 1.4, and typical values are listed in Tables 1.1 and 1.2. Poisson’s ratio nf is assumed to be a constant, for ZrO2=Ti-6Al-4V plate n ¼ 0.3, and for Si3N4=SUS304 one n ¼ 0.28. For the ﬁrst example, we consider the nonlinear free vibration of an isotropic square plate (a=b ¼ 1.0, b=h ¼ 10, and n ¼ 0.3) under different thermal loading conditions DT=Tcr ¼ 0, 0.25, 0.5, and 0.75, where Tcr ¼ 119.783= (a104) is the critical temperature of the plate (Bhimaraddi and Chandrashekhara 1993). The frequency parameter V ¼ vL (a2 =h)[r(1 n2 )=E]1=2 and nonlinear to linear frequency ratio vNL=vL are calculated and compared in Table 4.1 with the results of Bhimaraddi and Chandrashekhara (1993) based on the CPT, FSDPT, and HSDPT. For the second example, we consider the free vibration of an FGM square plate made of aluminum oxide and Ti-6Al-4V. The top surface is ceramicrich, whereas the bottom surface is metal-rich. The material properties, as given in He et al. (2001), are Em ¼ 105.7 GPa, nm ¼ 0.2981, rm ¼ 4429 kg m3

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 4.1 Comparison of Natural Frequency V and Nonlinear to Linear Frequency Ratios for an Isotropic Square Plate under Different Thermal Loading Conditions (a=b ¼ 1.0, b=h ¼ 10, and n ¼ 0.3) W max=h DT=Tcr 0.25

0.5

0.75

a

Sources a

V

0.0

0.2

0.4

0.6

0.8

1.0

HSDPT

4.7624

1.000

1.027

1.105

1.222

1.368

1.535

FSDPTa

4.7232

0.922

1.019

1.097

1.215

1.362

1.529

CPTa

4.9380

1.037

1.063

1.138

1.252

1.395

1.559

Present

4.7636

1.000

1.027

1.105

1.225

1.374

1.546

HSDPT

3.8884

1.000

1.041

1.153

1.318

1.517

1.739

FSDPT

3.8405

0.988

1.029

1.143

1.309

1.509

1.732

CPT Present

4.1017 3.8891

1.055 1.000

1.094 1.040

1.201 1.155

1.360 1.323

1.554 1.528

1.772 1.757

HSDPT

2.7495

1.000

1.080

1.287

1.569

1.893

2.242

FSDPT

2.6813

0.975

1.057

1.267

1.553

1.880

2.230

CPT

3.0437

1.107

1.180

1.372

1.640

1.953

2.293

Present

2.7492

1.000

1.080

1.291

1.582

1.916

2.275

HSDPT, FSDPT, and CPT results all from Bhimaraddi and Chandrashekhara (1993).

for Ti-6Al-4V; and Ec ¼ 320.24 GPa, nc ¼ 0.26, rc ¼ 3750 kg m3 for aluminum oxide. The FGM plate has a ¼ b ¼ 0.4 m and h ¼ 5 mm. Table 4.2 gives the comparison of natural frequency v L (in Hz) for the two special cases of isotropy, i.e., volume fraction index N ¼ 0 and 2000. The FEM results of He et al. (2001) based on the CPT and seminumerical results of Yang and Shen (2002) based on HSDPT are given for direct comparison. TABLE 4.2 Comparison of Natural Frequency vL (Hz) for Simply Supported FGM Plates for the Two Special Cases of Isotropy N¼0 Mode Sequence

N ¼ 2000

He et al. (2001)

Yang and Shen (2002)

Huang and Shen (2004)

He et al. (2001)

Yang and Shen (2002)

Huang and Shen (2004)

1

144.66

143.96

144.94

268.92

261.46

271.03

2

360.53

360.07

362.04

669.40

653.14

677.04

3

360.53

360.07

362.04

669.40

653.14

677.04

4 5

569.89 720.57

568.88 718.22

578.78 723.06

1052.49 1338.52

1044.31 1304.79

1082.38 1352.24

6

720.57

718.22

723.06

1338.52

1304.79

1352.24

7

919.74

916.40

939.19

1695.23

1694.98

1756.49

8

919.74

916.40

939.19

1695.23

1694.98

1756.49

9

1225.72

1207.09

1226.19

2280.95

2214.34

2294.47

10

1225.72

1207.09

1226.19

2280.95

2214.34

2294.47

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Nonlinear Vibration of Shear Deformable FGM Plates

111

We then examine the dynamic response of an FGM square plate subjected to a uniform sudden load with q0 ¼ 1.0 MPa in thermal environments. The FGM plate is made of aluminum and alumina. The side and thickness of the square plate are 200 and 10 mm, respectively. The top surface is ceramic-rich, whereas the bottom surface is metal-rich. The temperature is varied only in the thickness direction and determined by the steady-state heat conduction equation with the boundary conditions. A stress-free temperature T0 ¼ 08C was taken. The material properties adopted are Eb ¼ 70 GPa, nb ¼ 0.3, rb ¼ 2707 kg m3, ab ¼ 23.0 106 8C1, kb ¼ 204 W mK1, for aluminum; and Et ¼ 380 GPa, nt ¼ 0.3, rt ¼ 3800 kg m3, at ¼ 7.4 106 8C1, kt ¼ 10.4 W mK1, for alumina. The curves of central deﬂection as functions of time are plotted and compared in Figure 4.1 with the FEM results of Praveen and Reddy (1998) based on FSDPT. In Figure 4.1, dimensionless central deﬂection and time are deﬁned by W ¼ (Em h=q0 a2 ) and ~t ¼ t[Em =a2 rm ]1=2 , respectively. Note that in these three examples the material properties are assumed to be independent of temperature. Table 4.3 gives comparisons of frequency parameter for Si3N4=SUS304 square plates with temperature-dependent material properties under heat conduction. The results of Huang and Shen (2004) based on HSDPT, the FEM results of Sundararajan et al. (2005) based on the Mori–Tanaka scheme, and the differential quadrature method (DQM) results of Wu et al. (2007) are also given for direct comparison. These four comparisons show that the present results agree well with existing results, and can be used as benchmark for other numerical studies.

Dimensionless deflection

30 Ceramic, present Ceramic, Praveen and Reddy (1998) N = 0.5, present N = 0.5, Praveen and Reddy (1998)

25

20

15

10 0.0

b /h = 20 Tb = 20⬚C Tt = 300⬚C

2.5

5.0

7.5

10.0

12.5

Dimensionless time FIGURE 4.1 Comparison of central deﬂection versus time curves for an FGM square plate subjected to a suddenly applied uniform load and in thermal environments.

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112

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells TABLE 4.3 Comparisons of Frequency Parameter for Si3N4=SUS304 Square Plates under Heat Conduction Mode

N ¼ 0.0

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

(1, 1)

12.397

8.615

7.474

6.639

(1, 2)

29.083

20.215

17.607

15.762 23.786

TU ¼ 400 K, TL ¼ 300 K Huang and Shen (2004)

Sundararajan et al. (2005)

Wu et al. (2007)

TU ¼ 600 K, TL ¼ 300 K Huang and Shen (2004)

Sundararajan et al. (2005)

Wu et al. (2007)

(2, 2)

43.835

30.530

26.590

(1, 1)

12.311

8.276

7.302

6.572

(1, 2)

29.016

19.772

17.369

15.599

(2, 2) (1, 1)

44.094 12.353

30.184 8.513

26.506 7.439

23.787 6.678

(1, 2)

29.033

20.115

17.578

15.717

(2, 2)

43.775

30.226

26.510

23.699

(1, 1)

11.984

8.269

7.171

6.398

(1, 2)

28.504

19.783

17.213

15.384 23.327

(2, 2)

43.107

29.998

26.109

(1, 1)

11.888

7.943

6.989

6.269

(1, 2)

28.421

19.327

16.959

15.207

(2, 2)

43.343

29.629

25.997

23.303

(1, 1) (1, 2)

11.958 28.433

8.253 19.468

7.144 17.116

6.378 15.375

(2, 2)

43.003

29.886

25.994

23.315

Tables 4.4 and 4.5 show the effect of volume fraction index N on the natural frequency parameter of ZrO2=Ti-6Al-4V and Si3N4=SUS304 plates under three thermal loading conditions: case 1, TL ¼ 300 K, TU ¼ 300 K; case 2, TL ¼ 300 K, TU ¼ 400 K; and case 3, TL ¼ 300 K, TU ¼ 600 K. Temperature-dependent

TABLE 4.4

1=2 for ZrO2=Ti-6Al-4V Natural Frequency Parameter V ¼ v L (a2 =h) r0 (1 n2 )=E0 Square Plates in Thermal Environments Mode (1, 1)

(1, 2)

(2, 2)

(1, 3)

(2, 3)

ZrO2

8.273

19.261

28.962

34.873

43.070

0.5

7.139

16.643

25.048

30.174

37.288

1.0

6.657

15.514

23.345

28.120

34.747

2.0

6.286

14.625

21.978

26.454

32.659

Ti-6Al-4V

5.400

12.571

18.903

22.762

28.111

TL ¼ 300 K, TU ¼ 300 K

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Nonlinear Vibration of Shear Deformable FGM Plates

113

TABLE 4.4 (continued)

1=2 for ZrO2=Ti-6Al-4V Natural Frequency Parameter V ¼ v L (a2 =h) r0 (1 n2 )=E0 Square Plates in Thermal Environments Mode (1, 1)

(1, 2)

(2, 2)

(1, 3)

(2, 3)

TL ¼ 300 K, TU ¼ 400 K, temperature dependent ZrO2

7.868

18.659

28.203

34.015

42.045

0.5

6.876

16.264

24.578

29.651

36.664

1.0 2.0

6.437 6.101

15.202 14.372

22.956 21.653

27.696 26.113

34.236 32.239

Ti-6Al-4V

5.322

12.455

18.766

22.603

27.921

TL ¼ 300 K, TU ¼ 400 K, temperature independent ZrO2 0.5

8.122 7.154

19.193 16.644

28.986 25.136

34.958 30.136

43.190 37.476

1.0

6.592

15.531

23.442

28.273

34.936

2.0

6.238

14.655

22.078

26.605

32.840

Ti-6Al-4V

5.389

12.620

19.104

22.905

28.261

TL ¼ 300 K, TU ¼ 600 K, temperature dependent 6.685 16.986

26.073

31.567

39.212

0.5

6.123

15.169

23.166

28.041

34.789

1.0

5.819

14.287

21.768

26.342

32.660

2.0

5.612

13.611

20.652

24.961

30.904

Ti-6Al-4V

5.118

12.059

18.175

21.898

27.045

ZrO2

TL ¼ 300 K, TU ¼ 600 K, temperature independent ZrO2

7.686

18.749

28.527

34.472

42.713

0.5

6.776

16.367

24.859

30.044

37.201

1.0

6.362

15.308

23.216

28.036

34.714

2.0 Ti-6Al-4V

6.056 5.284

14.474 12.511

21.896 18.902

26.435 22.784

32.664 28.168

TABLE 4.5

1=2 Natural Frequency Parameter V ¼ v L (a2 =h) r0 (1 n2 )=E0 for Si3N4=SUS304 Square Plates in Thermal Environments Mode (1, 1)

(1, 2)

(2, 2)

(1, 3)

(2, 3)

TL ¼ 300 K, TU ¼ 300 K Si3N4

12.495

29.131

43.845

52.822

65.281

0.5

8.675

20.262

30.359

36.819

45.546

1.0

7.555

17.649

26.606

32.081

39.692

2.0

6.777

15.809

23.806

28.687

35.466

SUS304

5.405

12.602

18.967

22.850

28.239 (continued)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 4.5 (continued)

1=2 for Si3N4=SUS304 Natural Frequency Parameter V ¼ v L (a2 =h) r0 (1 n2 )=E0 Square Plates in Thermal Environments Mode (1, 1)

(1, 2)

(2, 2)

(1, 3)

(2, 3)

TL ¼ 300 K, TU ¼ 400 K, temperature dependent Si3N4

12.397

29.083

43.835

52.822

65.310

0.5

8.615

20.215

30.530

36.824

45.575

1.0 2.0

7.474 6.693

17.607 15.762

26.590 23.786

32.088 28.686

39.721 35.491

SUS304

5.311

12.539

18.959

22.828

28.246

TL ¼ 300 K, TU ¼ 400 K, temperature independent Si3N4 0.5

12.382 8.641

29.243 20.316

44.072 30.682

53.105 37.007

65.559 45.802

1.0

7.514

17.694

26.717

32.242

39.908

2.0

6.728

15.836

23.893

28.816

35.648

SUS304

5.335

12.587

19.008

22.908

28.344

TL ¼ 300 K, TU ¼ 600 K, temperature dependent 11.984 28.504

43.107

51.998

64.358

0.5

8.269

19.783

29.998

36.239

44.901

1.0

7.171

17.213

26.109

31.557

39.114

2.0

6.398

15.384

23.327

28.185

34.918

SUS304

4.971

12.089

18.392

22.221

27.557

Si3N4

TL ¼ 300 K, TU ¼ 600 K, temperature independent Si3N4

12.213

28.976

43.797

52.821

65.365

0.5

8.425

20.099

30.458

36.781

45.572

1.0

7.305

17.486

26.506

31.970

39.692

2.0 SUS304

6.523 5.104

15.632 12.342

23.685 18.763

28.609 22.658

35.436 28.084

and temperature-independent material properties (values at ﬁxed temperature 300 K) are both taken into account. In these two tables V ¼ vL (a2 =h)[r0 (1 n2 )=E0 ]1=2 , where E0 and r0 are the reference values of Em and rm at T0 ¼ 300 K. Then Tables 4.6 and 4.7 show, respectively, the effects of volume fraction index N and temperature ﬁeld on the nonlinear to linear frequency ratios vNL=vL of the same two FGM plates. It can be seen that the natural frequency of the FGM plate decreases with the increase of volume fraction index N, but it has a small effect on the nonlinear to linear frequency ratios. On the other hand, the temperature rise decreases the natural frequencies but increases the nonlinear to linear frequency ratios. The results show that the FGM plate will have lower natural frequency and slightly higher nonlinear to linear frequency ratios when the temperaturedependent material properties are taken into account.

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Nonlinear Vibration of Shear Deformable FGM Plates

115

TABLE 4.6 Effect of Volume Fraction Index N on the Nonlinear to Linear Frequency Ratio vNL=vL of FGM Square Plates in Thermal Environments (TL ¼ 300 K, TU ¼ 400 K) W max=h 0.0

0.2

0.4

0.6

0.8

1.0

ZrO2

1.000

1.023

1.087

1.186

1.312

1.461

0.5

1.000

1.023

1.087

1.186

1.312

1.460

1.0 2.0

1.000 1.000

1.022 1.022

1.086 1.084

1.183 1.179

1.310 1.302

1.455 1.444

Ti-6Al-4V

1.000

1.022

1.083

1.177

1.300

1.440

Si3N4 0.5

1.000 1.000

1.022 1.022

1.084 1.084

1.181 1.181

1.303 1.302

1.446 1.444

1.0

1.000

1.022

1.084

1.180

1.301

1.442

2.0

1.000

1.022

1.082

1.176

1.299

1.440

SUS304

1.000

1.022

1.082

1.172

1.296

1.438

ZrO2=Ti-6Al-4V

Si3N4=SUS304

TABLE 4.7 Effect of Temperature Field on the Nonlinear to Linear Frequency Ratio vNL=vL of FGM Square Plates (N ¼ 2.0) W max=h 0.0

0.2

0.4

0.6

0.8

1.0

TL ¼ 300 K, TU ¼ 300 K

1.000

1.021

1.082

1.176

1.296

1.436

TL ¼ 300 K, TU ¼ 400 K Temperature dependent

1.000

1.022

1.084

1.179

1.302

1.444

Temperature independent

1.000

1.022

1.083

1.178

1.300

1.441

TL ¼ 300 K, TU ¼ 600 K Temperature dependent

1.000

1.024

1.091

1.194

1.325

1.477

Temperature independent

1.000

1.023

1.087

1.183

1.314

1.462

1.000

1.021

1.081

1.174

1.293

1.432

Temperature dependent

1.000

1.022

1.082

1.176

1.299

1.440

Temperature independent

1.000

1.021

1.082

1.175

1.255

1.437

Temperature dependent

1.000

1.023

1.088

1.188

1.315

1.463

Temperature independent

1.000

1.023

1.087

1.187

1.313

1.460

ZrO2=Ti-6Al-4V

Si3N4=SUS304 TL ¼ 300 K, TU ¼ 300 K TL ¼ 300 K, TU ¼ 400 K

TL ¼ 300 K, TU ¼ 600 K

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116

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Figures 4.2 and 4.3 show, respectively, the effect of volume fraction index N on the dynamic response of ZrO2=Ti-6Al-4V and Si3N4=SUS304 plates under thermal environmental condition TL ¼ 300 K and TU ¼ 400 K. It can 2 TL = 300 K, TU = 400 K

Central deflection (mm)

0

−2

−4

−6

0.0

ZrO2;

N = 0.5;

N = 2.0;

Ti-6Al-4V

0.2

(a)

Central bending moment (kN m m–1)

0.6

0.8

Time (ms) 0

TL = 300 K, TU = 400 K

−10

−20

−30

−40

−50 0.0 (b)

0.4

N = 1.0;

ZrO2;

N =0.5;

N =2.0;

Ti-6Al-4V

0.2

0.4

N =1.0;

0.6

0.8

Time (ms)

FIGURE 4.2 Effect of volume fraction index N on the dynamic response of ZrO2=Ti-6Al-4V square plate subjected to a suddenly applied uniform load and in thermal environments: (a) central deﬂection versus time; (b) bending moment versus time.

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Nonlinear Vibration of Shear Deformable FGM Plates

117

2 N =0.5;

Si3N4; N =1.0;

Central deflection (mm)

1

N =2.0;

SUS304

0

−1

−2 TL = 300 K, TU = 400 K −3 0.0

0.2

(a)

0.4

0.6

0.8

0.6

0.8

Time (ms)

Central bending moment (kN m m−1)

−10 Si3N4;

N =0.5;

N =1.0;

N =2.0;

SUS304 −20

−30

TL = 300 K, TU = 400 K −40 0.0 (b)

0.2

0.4 Time (ms)

FIGURE 4.3 Effect of volume fraction index N on the dynamic response of Si3N4=SUS304 square plate subjected to a suddenly applied uniform load and in thermal environments: (a) central deﬂection versus time; (b) bending moment versus time.

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118

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

be seen that the plate deﬂections are increased by increasing the volume fraction index N. The bending moment is decreased for the ZrO2=Ti-6Al-4V plate, but it is increased for the Si3N4=SUS304 plate when the volume fraction index N is increased. Figures 4.4 and 4.5 show, respectively, the effect of thermal environmental conditions on the dynamic response of ZrO2=Ti-6Al-4V and Si3N4=SUS304 plates with N ¼ 2.0. The results show that both central deﬂections and bending moments are increased with the increase in temperature. It is also seen that the greater the temperature rise is, the greater will be the thermally induced initial bending moments. It is appreciated that in Figures 4.2 through 4.5 the deﬂection mode (m, n) ¼ (1, 1) was used and in Equation 4.19 k and l are taken as 1, 3, and 5, and in Equations 4.41 and 4.42 Dt ¼ 2 ms is taken as the time step for Runge– Kutta iteration method. The dynamic load is assumed to be a suddenly applied uniform load with q0 ¼ 50 MPa.

4.3

Nonlinear Vibration of FGM Plates with Piezoelectric Actuators in Thermal Environments

We now consider the nonlinear free and forced vibration of FGM hybrid laminated plates. The plate is assumed to be made of a substrate FGM layer with surface-bonded piezoelectric layers. The substrate FGM layer is made of the combined ceramic and metallic materials with continuously varying mix-ratios comprising ceramic and metal. The length, width, and total thickness of the hybrid laminated plate are a, b, and h. The thickness of the FGM layer is hf, while the thickness of the piezoelectric layer is hp. We assume that the material composition varies smoothly from the upper to the lower surface of the FGM layer, such that the upper surface (Z ¼ h2) of the FGM layer is ceramic-rich, and the lower surface (Z ¼ h1) is metal-rich. As in the case of Section 4.2, we assume the effective Young’s modulus Ef and thermal expansion coefﬁcient af of the FGM layer are temperature dependent, whereas the thermal conductivity kf and mass density rf are independent to the temperature. Poisson’s ratio nf depends weakly on temperature change and is assumed to be a constant. We assume that the volume fraction Vc follows a simple power law. Now Equation 4.2 may be rewritten as Ef (Z, T) ¼ [Ec (T) Em (T)]

Z h1 h2 h1

N þ Em (T)

(4:43a)

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Nonlinear Vibration of Shear Deformable FGM Plates

119

4 TL = 300 K, TU = 300 K

N = 2.0

TL = 300 K, TU = 400 K TL = 300 K, TU = 600 K

Central deflection (mm)

2

0

−2

−4

0.0

0.2

(a)

0.4 Time (ms)

0.6

0.8

20

Central bending moment (kN m m−1)

0 –20 –40 TL = 300 K, TU = 300 K –60 –80

TL = 300 K, TU = 400 K TL = 300 K, TU = 600 K

–100 –120 0.0 (b)

0.2

0.4

0.6

0.8

Time (ms)

FIGURE 4.4 Effect of temperature ﬁeld on the dynamic response of ZrO2=Ti-6Al-4V square plate subjected to a suddenly applied uniform load: (a) central deﬂection versus time; (b) bending moment versus time.

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120

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 3 TL = 300 K, TU = 300 K

N =2.0

2

TL = 300 K, TU = 400 K

Central deflection (mm)

TL = 300 K, TU = 600 K 1

0

−1

−2

−3 0.0

0.2

(a)

0.4

0.6

0.8

Time (ms) 20

Central bending moment (kN m m–1)

N =2.0 0

−20

−40

TL = 300 K, TU = 400 K −60

−80 0.0 (b)

TL = 300 K, TU = 300 K

TL = 300 K, TU = 600 K

0.2

0.4

0.6

0.8

Time (ms)

FIGURE 4.5 Effect of temperature ﬁeld on the dynamic response of Si3N4=SUS304 square plate subjected to a suddenly applied uniform load: (a) central deﬂection versus time; (b) bending moment versus time.

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C004 Final Proof page 121 9.12.2008 4:38pm Compositor Name: VBalamugundan

Nonlinear Vibration of Shear Deformable FGM Plates Z h1 N af (Z, T) ¼ [ac (T) am (T)] þ am (T) h2 h1 Z h1 N þ rm rf (Z) ¼ (rc rm ) h2 h1 Z h1 N þ km kf (Z) ¼ (kc km ) h 2 h1

121

(4:43b) (4:43c) (4:43d)

It is evident that when Z ¼ h1, Ef ¼ Em(Tm) and af ¼ am(Tm), and when Z ¼ h2, Ef ¼ Ec(Tc) and af ¼ ac(Tc). Furthermore, Ef and af are both temperature- and position-dependent. The temperature ﬁeld is assumed to be the same as in Section 4.2. Then the temperature distribution along the thickness can be obtained by solving a steady-state heat transfer equation:

d dT k(Z) ¼0 dZ dZ

(4:44)

where 8 > < kp k(Z) ¼ kf (Z) > : kp 8 ~ > < Tp (Z) T(Z) ¼ Tf (Z) > : Tp (Z)

(h0 < Z < h1 ) (h1 < Z < h2 ) (h2 < Z < h3 )

(4:45a)

(h0 Z h1 ) (h1 Z h2 ) (h2 Z h3 )

(4:45b)

where kp is the thermal conductivity of the piezoelectric layer. Equation 4.44 is solved by imposing the boundary conditions T ¼ TU at Z ¼ h3 and T ¼ TL at Z ¼ h0, and the continuity conditions

~ p (Z) dT kp dZ

~ p (h1 ) ¼ Tf (h1 ) ¼ Tm , Tp (h2 ) ¼ Tf (h2 ) ¼ Tc T dTp (Z) dTf (Z) dTf (Z) ¼ km , kp ¼ kc dZ Z¼h2 dZ Z¼h1 dZ Z¼h2

(4:46a) (4:46b)

Z¼h1

The solution of Equations 4.44 through 4.46, by means of polynomial series, is ~ p (Z) ¼ 1 [(TL h1 Tm h0 ) þ (Tm TL )Z] T hp

(4:47a)

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122

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells Tf (Z) ¼ Tm þ (Tc Tm )h(Z) Tp (Z) ¼

1 [(Tc h3 TU h2 ) þ (TU Tc )Z] hp

(4:47b) (4:47c)

where 1 h(Z) ¼ C

"

Z h1 h2 h1

kmc Z h1 Nþ1 k2mc Z h1 2Nþ1 þ (N þ 1)kc h2 h1 (2N þ 1)k2c h2 h1

k3mc Z h1 3Nþ1 k4mc Z h1 4Nþ1 þ (3N þ 1)k3c h2 h1 (4N þ 1)k4c h2 h1 # k5mc Z h1 5Nþ1 (5N þ 1)k5c h2 h1

C¼1

kmc k2mc k3mc k4mc k5mc þ þ 2 3 4 (N þ 1)kc (2N þ 1)kc (3N þ 1)kc (4N þ 1)kc (5N þ 1)k5c G¼1

kmc k2mc k3mc k4mc k5mc þ 2 3 þ 4 5 kc kc kc kc kc

(4:48a)

(4:48b) (4:48c)

where kmc ¼ km kc, and Tc ¼

(1=hf C)(kc GTL þ km TU ) þ (kp =hp )TU (1=hf C)(kc G þ km ) þ (1=hp )kp

(4:49a)

Tm ¼

(1=hf C)(kc GTL þ km TU ) þ (kp =hp )TL (1=hf C)(kc G þ km ) þ (1=hp )kp

(4:49b)

The plate is assumed to be geometrically perfect, and is subjected to a transverse dynamic load q(X, Y, t) in thermal environments. Hence, the general von Kármán-type equations can be written in a similar form as expressed by Equations 1.33 through 1.36, just necessary to replace N T, MT, ST, PT with N p, Mp, Sp, Pp, and these equivalent thermopiezoelectric loads are deﬁned by Equations 3.6 through 3.9. Introducing dimensionless quantities (Equations 2.8 and 4.6), and * D22 * ]1=2 , (g T1 , gT2 ) ¼ (ATx , ATy )a2 =p2 [D11

* D22 * ]1=2 (gP1 , g P2 ) ¼ (BEx , BEy )a2 =p2 [D11

* (g T3 , g T4 , gT6 , gT7 ) ¼ (DTx , DTy , FTx , FTy )a2 =p2 h2 D11 (gP3 , g P4 , g P6 , gP7 ) ¼

* (DEx , DEy , FEx , FEy )a2 =p2 h2 D11

(4:50)

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Nonlinear Vibration of Shear Deformable FGM Plates

123

where BEx ¼BEy , DEx ¼DEy , and FEx ¼FEy are deﬁned by "

BEx

DEx

FEx

BEy

DEy

FEy

# DV ¼

# hk " X ð Bx

k

hk1

By

k

1, Z, Z3

Vk dZ hk

(4:51)

The nonlinear motion equations can then be written in similar form as L11 (W) L12 (Cx ) L13 (Cy ) þ g 14 L14 (F) L16 (MP ) ! €y €x @ C @ C 2 € þg þb ¼ g14 b L(W, F) þ L17 (W) þ lq 80 @x @y

(4:52)

1 L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) g 24 L24 (W) ¼ g 24 b2 L(W, W) (4:53) 2 € @W € x (4:54) þ g 10 C L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) L36 (SP ) ¼ g 90 @x € @W € y (4:55) þ g 10 C L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) L46 (SP ) ¼ g 90 b @y and the boundary conditions can be written in dimensionless form as x ¼ 0, p: W ¼ Cy ¼ 0 ðp ðp g224 b2 0 0

(4:56a)

@Cy @2F @2F @Cx þ g 233 b g5 2 þ g 24 g 511 @x @y @y2 @x

2 @2W 1 @W 2 2@ W g 24 þ g244 b g24 g 611 @x2 @y2 2 @x 2 2 þ (g 24 gT1 g5 gT2 )T1 þ (g24 g P1 g5 gP2 )DV dx dy ¼ 0

(4:56b)

y ¼ 0, p: W ¼ Cx ¼ 0

(4:56c)

ðp ðp 2 2 @Cy @ F @Cx 2@ F þ g 522 b g5 b þ g 24 g 220 @x @y @x2 @y2 0 0

2 2 @2W 1 2@ W 2 @W g24 b þ g 622 b g 24 g 240 @x2 @y2 2 @y þ (g T2 g 5 g T1 )T1 þ (g P2 g5 g P1 )DV dx dy ¼ 0

(4:56d)

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124

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

All the necessary steps of the solution methodology are described in Section 4.2, and the solutions are not repeated herein for convenience. We ﬁrst examine the free vibration of an FGM square plate with symmetrically fully covered G-1195N piezoelectric layers. The substrate FGM plate is made of aluminum oxide and Ti-6Al-4V. The material properties adopted are Ec ¼ 320.24 GPa, nc ¼ 0.26, rc ¼ 3750 kg m3, for aluminum oxide; Em ¼ 105.70 GPa, nm ¼ 0.2981, rm ¼ 4429 kg m3, for Ti-6Al-4V; Ep ¼ 63.0 GPa, np ¼ 0.3, rp ¼ 7600 kg m3, d31 ¼ d32 ¼ 254 1012 mV1. The side and thickness of the substrate FGM square plate are 400 and 5 mm, and the thickness of each piezoelectric layer is 0.1 mm. The initial 10 frequencies of the plate as a function of the volume fraction index N are listed in Table 4.8 and compared with the FEM results of He et al. (2001) based on classical laminated plate theory (CLPT). Again, good agreement can be seen. Note that in this example the material properties are assumed to be independent of temperature. In this section, two types of the hybrid FGM plate are considered. The ﬁrst hybrid FGM plate has fully covered piezoelectric actuators on the top surface (referred to as P=FGM), and the second has two piezoelectric layers symmetrically bonded to the top and bottom surfaces (referred to as P=FGM=P). Silicon nitride (Si3N4) and stainless steel (SUS304) are chosen to be the TABLE 4.8 Comparison of Natural Frequency vL (Hz) for FGM Plates with Piezoelectric Actuator Bonded on the Top and Bottom Surfaces Mode 1 2 3 4 5 6 7 8 9 10

Method

N¼0

N ¼ 0.5

N¼1

N¼5

N ¼ 15

N ¼ 100

N ¼ 1000 261.73

He et al. (2001)

144.25

185.45

198.92

230.46

247.30

259.35

Present

143.25

184.73

198.78

229.47

246.86

258.78

260.84

He et al. (2001)

359.00

462.65

495.62

573.82

615.58

645.55

651.49

Present

358.87

461.02

494.65

571.87

613.95

643.92

649.83

He et al. (2001) Present

359.00 358.87

462.47 461.02

495.62 494.65

573.82 571.87

615.58 613.95

645.55 643.92

651.49 649.83 1024.28

He et al. (2001)

564.10

731.12

778.94

902.04

967.78

1014.94

Present

563.42

727.98

778.61

899.91

964.31

1012.54

1023.72

He et al. (2001)

717.80

925.45

993.11

1148.12

1231.00

1290.78

1302.64

Present

717.65

922.83

992.87

1146.87

1229.44

1288.73

1301.34

He et al. (2001)

717.80

925.45

993.11

1148.12

1231.00

1290.78

1302.64

Present

717.65

922.83

992.87

1146.87

1229.44

1288.73

1301.34

He et al. (2001) Present

908.25 907.87

1180.93 1177.34

1255.98 1223.36

1453.32 1451.66

1558.77 1557.12

1634.65 1632.18

1649.70 1648.56

He et al. (2001)

908.25

1180.93

1255.98

1453.32

1558.77

1634.65

1649.70

Present

907.87

1177.34

1223.36

1451.66

1557.12

1632.18

1648.56

1223.14

1576.91

1697.15

1958.17

2097.91

2199.46

2219.67

He et al. (2001) Present

1219.32

1571.65

1695.17

1956.79

2095.67

2197.47

2217.94

He et al. (2001)

1223.14

1576.91

1697.15

1958.17

2097.91

2199.46

2219.67

Present

1219.32

1571.65

1695.17

1956.79

2095.67

2197.47

2217.94

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Nonlinear Vibration of Shear Deformable FGM Plates

125

constituent materials of the substrate FGM layer. The mass density rf, Poisson’s ratio nf, and thermal conductivity kf are 2370 kg m3, 0.24, 9.19 W mK1 for Si3N4, and 8166 kg m3, 0.33, 12.04 W mK1 for SUS304. Young’s modulus Ef and thermal expansion coefﬁcient af for these two constituent materials are listed in Tables 1.1 and 1.2. PZT-5A is selected for the piezoelectric layers. The material properties of which, as linear functions of temperature of Equation 3.35, are E110 ¼ E220 ¼ 63 GPa, G120 ¼ G130 ¼ G230 ¼ 24.2 GPa, a110 ¼ a220 ¼ 0.9 106 K1, rp ¼ 7600 kg m3, np ¼ 0.3, kp ¼ 2.1 W mK1, and d31 ¼ d32 ¼ 2.54 1010 mV1, and E111 ¼ 0.0005, E221 ¼ G121 ¼ G131 ¼ G231 ¼ 0.0002, a111 ¼ a221 ¼ 0.0005. The side of the hybrid FGM plate is a ¼ b ¼ 24 mm. The thickness of the substrate FGM 1layer hf ¼ 1.0 mm, whereas the thickness of each piezoelectric layer hp ¼ 0.1 mm. Tables 4.9 and 4.10 present the natural frequency parameter V ¼ vL (a2 =hf )[r0 =E0 ]1=2 of these two types of the FGM hybrid plate with different TABLE 4.9 Natural Frequency Parameter V ¼ v L (a2 =hf )[r0 =E0 ]1=2 for the Hybrid (P=FGM) Plates under Different Sets of Thermal and Electric Loading Conditions VU ¼ VL ¼

Si3N4 (N ¼ 0)

N ¼ 0.5

N ¼ 2.0

N ¼ 4.0

SUS304 (N ¼ 1)

200 V 0V

10.726 10.704

7.885 7.868

6.334 6.320

5.920 5.906

5.194 5.179

þ200 V

10.682

7.852

6.306

5.892

5.164

200 V

10.149

7.380

5.852

5.427

4.668

0V

10.134

7.371

5.846

5.422

4.665

þ200 V

10.119

7.363

5.841

5.418

4.663

200 V

9.237

6.667

5.277

4.903

4.324

0V

9.248

6.685

5.299

4.926

4.352

þ200 V

9.260

6.704

5.322

4.950

4.378

(P=FGM), TID TU ¼ 300 K, TL ¼ 300 K TU ¼ 400 K, TL ¼ 300 K TU ¼ 600 K, TL ¼ 300 K (P=FGM), TD-F TU ¼ 400 K, TL ¼ 300 K

200 V

10.099

7.346

5.824

5.402

4.649

0V

10.084

7.336

5.819

5.397

4.646 4.645

þ200 V

10.070

7.328

5.814

5.393

TU ¼ 600 K,

200 V

9.093

6.592

5.242

4.882

3.432

TL ¼ 300 K

0V 200 V

9.109 9.125

6.614 6.636

5.266 5.291

4.907 4.932

3.277 3.159

(P=FGM), TD TU ¼ 400 K, TL ¼ 300 K TU ¼ 600 K, TL ¼ 300 K

200 V

10.080

7.331

5.812

5.390

4.635

0V

10.066

7.322

5.807

5.385

4.633

200 V

10.052

7.315

5.803

5.382

4.632

200 V 0V

9.042 9.055

6.557 6.576

5.216 5.238

4.858 4.881

3.183 3.089

200 V

9.070

6.595

5.259

4.902

3.010

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126

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 4.10 Natural Frequency Parameter V ¼ v L (a2 =hf )[r0 =E0 ]1=2 for the Hybrid (P=FGM=P) Plates under Different Sets of Thermal and Electric Loading Conditions Si3N4 (N ¼ 0)

N ¼ 0.5

N ¼ 2.0

N ¼ 4.0

SUS304 (N ¼ 1)

200 V

9.121

7.032

5.794

5.446

4.810

0V

9.085

7.000

5.766

5.418

4.782

þ200 V

9.050

6.969

5.738

5.391

4.755

TU ¼ 400 K, TL ¼ 300 K

200 V 0V

8.435 8.397

6.344 6.310

5.102 5.070

4.740 4.709

4.058 4.026

þ200 V

8.358

6.276

5.038

4.678

3.995

TU ¼ 600 K,

200 V

7.124

5.004

3.886

3.622

1.521

VU ¼ VL ¼ (P=FGM=P), TID TU ¼ 300 K, TL ¼ 300 K

TL ¼ 300 K

0V

7.085

4.971

3.860

3.601

1.424

þ200 V

7.046

4.938

3.837

3.581

1.323

200 V

8.372

6.295

5.059

4.698

4.022

0V

8.333

6.260

5.027

4.668

3.990

þ200 V

8.294

6.226

4.996

4.636

3.959

200 V

6.900

4.854

3.816

2.511

1.287

0V

6.863

4.825

3.797

2.370

1.172

200 V

6.826

4.795

3.778

2.246

1.045

(P=FGM=P), TD-F TU ¼ 400 K, TL ¼ 300 K TU ¼ 600 K, TL ¼ 300 K

(P=FGM=P), TD TU ¼ 400 K,

200 V

8.340

6.266

5.033

4.673

3.996

TL ¼ 300 K

0V 200 V

8.303 8.266

6.233 6.200

5.002 4.972

4.643 4.613

3.966 3.936

TU ¼ 600 K,

200 V

6.806

4.777

3.763

2.190

0.976

0V

6.775

4.752

3.747

2.094

0.847

200 V

6.744

4.729

3.731

2.003

0.695

TL ¼ 300 K

values of the volume fraction index N ( ¼ 0.0, 0.5, 2.0, 4.0, and 1) under different sets of thermal and electric loading conditions. Here, E0 and r0 are the reference values of SUS304 at the room temperature (T0 ¼ 300 K). TD represents material properties for both substrate FGM layer and piezoelectric layers are temperature dependent. TD-F represents material properties of substrate FGM layer are temperature dependent but material properties of piezoelectric layers are temperature independent, i.e., E111 ¼ E221 ¼ G121 ¼ G131 ¼ G231 ¼ a111 ¼ a221 ¼ 0 in Equation 3.35. TID represents material properties for both piezoelectric layers and substrate FGM layer are temperature independent, i.e., in a ﬁxed temperature T0 ¼ 300 K for FGM layer, as previously used in Yang and Shen (2001). Six different applied voltages: VU ¼ 200 V, VU ¼ 0 V, VU ¼ 200 V, and VL ¼ VU ¼ 200 V, VL ¼ VU ¼ 0 V, VL ¼ VU ¼ 200 V are used, where subscripts ‘‘L’’ and ‘‘U’’ imply the low and

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Nonlinear Vibration of Shear Deformable FGM Plates

127

upper piezoelectric layer. It can be seen that the natural frequency of these two plates is decreased by increasing temperature and volume fraction index N. The plus voltage decreases, but the minus voltage increases the plate natural frequency. Tables 4.11 through 4.13 show, respectively, the effect of volume fraction index N, control voltage, and temperature ﬁeld on the nonlinear to linear TABLE 4.11 Effect of Volume Fraction Index N on Nonlinear to Linear Frequency Ratio vNL =vL for the Hybrid FGM Plates in Thermal Environments (TL ¼ 300 K, TU ¼ 400 K) W max =h 0.2

0.4

0.6

0.8

1.0

(P=FGM) (VU ¼ þ200 V), TID 1.000 Si3N4

0.0

1.021

1.081

1.175

1.294

1.434

0.5

1.000

1.022

1.084

1.182

1.305

1.449

2.0

1.000

1.022

1.084

1.180

1.303

1.446

4.0

1.000

1.021

1.082

1.178

1.299

1.440

SUS304

1.000

1.022

1.087

1.188

1.315

1.463

(P=FGM) (VU ¼ þ200 V), TD-F 1.000 1.021 Si3N4

1.081

1.175

1.296

1.435

0.5

1.000

1.022

1.085

1.182

1.307

1.451

2.0

1.000

1.022

1.084

1.181

1.305

1.448

4.0

1.000

1.021

1.083

1.179

1.301

1.442

SUS304

1.000

1.023

1.088

1.189

1.317

1.466

(P=FGM) (VU ¼ þ200 V), TD 1.000 Si3N4

1.021

1.082

1.176

1.296

1.436

0.5

1.000

1.022

1.085

1.183

1.307

1.452

2.0

1.000

1.022

1.084

1.182

1.306

1.449

4.0

1.000

1.021

1.083

1.179

1.301

1.444

SUS304

1.000

1.023

1.088

1.190

1.318

1.467

(P=FGM=P) (VL ¼ VU ¼ þ200 V), TID Si3N4

1.000

1.021

1.082

1.177

1.298

1.439

0.5 2.0

1.000 1.000

1.022 1.022

1.087 1.088

1.186 1.188

1.313 1.315

1.460 1.463

4.0

1.000

1.022

1.087

1.187

1.314

1.460

SUS304

1.000

1.025

1.096

1.205

1.343

1.502

(P=FGM=P) (VL ¼ VU ¼ þ200 V), TD-F 1.000 1.021 Si3N4

1.082

1.177

1.299

1.439

0.5

1.000

1.023

1.087

1.187

1.315

1.462

2.0

1.000

1.023

1.088

1.189

1.318

1.467

4.0

1.000

1.023

1.088

1.188

1.316

1.465

SUS304

1.000

1.025

1.096

1.205

1.343

1.502 (continued)

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TABLE 4.11 (continued) Effect of Volume Fraction Index N on Nonlinear to Linear Frequency Ratio vNL =vL for the Hybrid FGM Plates in Thermal Environments (TL ¼ 300 K, TU ¼ 400 K) W max =h 0.0

0.2

0.4

0.6

0.8

1.0

(P=FGM=P) (VL ¼ VU ¼ þ200 V), TD Si3N4

1.000

1.021

1.083

1.178

1.300

1.441

0.5

1.000

1.023

1.088

1.188

1.316

1.464

2.0 4.0

1.000 1.000

1.023 1.023

1.089 1.088

1.190 1.190

1.320 1.318

1.470 1.468

SUS304

1.000

1.025

1.097

1.207

1.346

1.506

TABLE 4.12 Effect of Temperature Field on Nonlinear to Linear Frequency Ratio vNL =vL for the Hybrid FGM Plates (N ¼ 0.5) W max =h 0.2

0.4

0.6

0.8

1.0

(P=FGM) (VU ¼ þ200 V), TID 1.000 TL ¼ 300 K, TU ¼ 300 K

0.0

1.019

1.075

1.161

1.272

1.402

TL ¼ 300 K, TU ¼ 400 K

1.000

1.022

1.084

1.182

1.306

1.450

TL ¼ 300 K, TU ¼ 600 K

1.000

1.026

1.099

1.211

1.352

1.515

(P=FGM) (VU ¼ þ200 V), TD-F 1.000 TL ¼ 300 K, TU ¼ 400 K

1.022

1.085

1.182

1.307

1.451

TL ¼ 300 K, TU ¼ 600 K

1.000

1.025

1.096

1.205

1.343

1.502

(P=FGM) (VU ¼ þ200 V), TD 1.000 TL ¼ 300 K, TU ¼ 400 K

1.022

1.085

1.183

1.307

1.452

TL ¼ 300 K, TU ¼ 600 K

1.025

1.096

1.206

1.345

1.504

1.000

(P=FGM==P) (VU ¼ VL ¼ þ200 V), TID TL ¼ 300 K, TU ¼ 300 K

1.000

1.018

1.071

1.153

1.259

1.384

TL ¼ 300 K, TU ¼ 400 K

1.000

1.022

1.087

1.186

1.313

1.460

TL ¼ 300 K, TU ¼ 600 K

1.000

1.035

1.134

1.282

1.464

1.670

(P=FGM==P) (VU ¼ VL ¼ þ200 V), TD-F TL ¼ 300 K, TU ¼ 400 K

1.000

1.023

1.087

1.187

1.315

1.462

TL ¼ 300 K, TU ¼ 600 K

1.000

1.034

1.128

1.271

1.447

1.645

(P=FGM==P) (VU ¼ VL ¼ þ200 V), TD TL ¼ 300 K, TU ¼ 400 K

1.000

1.023

1.088

1.188

1.316

1.464

TL ¼ 300 K, TU ¼ 600 K

1.000

1.033

1.128

1.270

1.445

1.643

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Nonlinear Vibration of Shear Deformable FGM Plates

129

TABLE 4.13 Effect of Applied Voltage on Nonlinear to Linear Frequency Ratio vNL =vL for the Hybrid FGM Plates in Thermal Environments (TL ¼ 300 K, TU ¼ 400 K, N ¼ 2.0) W max =h 0.0

0.2

0.4

0.6

0.8

1.0

VU ¼ 200 V

1.000

1.022

1.084

1.180

1.302

1.445

VU ¼ 0 V

1.000

1.022

1.084

1.180

1.303

1.446

VU ¼ þ200 V

1.000

1.022

1.084

1.180

1.304

1.447

VU ¼ 200 V VU ¼ 0 V

1.000 1.000

1.022 1.022

1.084 1.084

1.180 1.181

1.304 1.304

1.446 1.447

VU ¼ þ200 V

1.000

1.022

1.084

1.181

1.305

1.448

(P=FGM), TD VU ¼ 200 V

1.000

1.022

1.084

1.181

1.304

1.448

VU ¼ 0 V

1.000

1.022

1.084

1.181

1.305

1.448

VU ¼ þ200 V

1.000

1.022

1.084

1.182

1.306

1.450

VU ¼ VL ¼ 200 V

1.000

1.022

1.085

1.183

1.308

1.454

VU ¼ V L ¼ 0 V

1.000

1.022

1.086

1.186

1.312

1.458

VU ¼ VL ¼ þ200 V

1.000

1.022

1.087

1.188

1.315

1.463

VU ¼ VL ¼ 200 V

1.000

1.022

1.086

1.185

1.311

1.457

VU ¼ V L ¼ 0 V VU ¼ VL ¼ þ200 V

1.000 1.000

1.022 1.023

1.087 1.088

1.187 1.189

1.314 1.318

1.462 1.467

VU ¼ VL ¼ 200 V

1.000

1.022

1.087

1.186

1.313

1.460

VU ¼ V L ¼ 0 V VU ¼ VL ¼ þ200 V

1.000 1.000

1.023 1.023

1.088 1.089

1.188 1.190

1.316 1.320

1.465 1.470

(P=FGM), TID

(P=FGM), TD-F

(P=FGM=P), TID

(P=FGM=P), TD-F

(P=FGM=P), TD

frequency ratios vNL=vL of these two types of FGM hybrid plates. It can be seen that the ratios vNL=vL increase as the volume fraction index N or temperature increases. It is noted that the control voltage only has a small effect on the frequency ratios. It is found that the decrease of natural frequency is about 46% for the P=FGM plate, and about 43% for the P=FGM=P one, from N ¼ 0 to N ¼ 4, in thermal environmental condition TL ¼ 300 K, TU ¼ 400 K under TD-F and TD cases. It can also be seen that the natural frequency and the nonlinear to linear frequency ratio under TD-F and TD cases are very close. Figures 4.6 and 4.7 show the effect of temperature dependency and volume fraction index N ( ¼ 0, 0.5, 2.0, and 1) on the dynamic response of P=FGM

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130

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 0.4

Central deflection, W (mm)

0.3

VU = 200 V

N = 2.0 (TD)

N = 0.5 (TD)

TU = 400 K

N = 2.0 (TID)

N = 0.5 (TID)

TL = 300 K

0.2

0.1

0.0 0.0

0.1

0.2

0.3

Time, t (ms)

(a)

Bending moment, M (kN m m−1)

−15 N = 2.0 (TD)

N = 0.5 (TD)

TU = 400 K

N= 2.0 (TID)

N = 0.5 (TID)

TL = 300 K

−20

−25 0.0 (b)

VU = 200 V

0.1

0.2

0.3

Time, t (ms)

FIGURE 4.6 Effect of temperature dependency on the dynamic response of P=FGM plate subjected to a sudden load, control voltage and in thermal environments: (a) central deﬂection versus time; (b) central bending moment versus time.

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Nonlinear Vibration of Shear Deformable FGM Plates

131

Central deflection, W (mm)

0.6 VU = 200 V

Si3N4

N = 0.5

TU = 400 K

N = 2.0

SUS304

TL = 300 K 0.4

0.2

0.0 0.0

0.1

(a)

0.2

0.3

Time, t (ms)

Bending moment, Mx (kN m m–1)

−20

−22

−24

−26

VU = 200 V

Si3N4

N = 0.5

TU = 400 K

N = 2.0

SUS304

TL = 300 K −28 (b) −30

(b)

0.0

0.1

0.2

0.3

Time, t (ms)

FIGURE 4.7 Effect of volume fraction index N on the dynamic response of P=FGM plate subjected to a suddenly applied uniform load, control voltage and in thermal environments: (a) central deﬂection versus time; (b) central bending moment versus time.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

plates subjected to a sudden applied load with q0 ¼ 2.0 MPa, under electric loading condition VU ¼ 200 V and in thermal environmental condition TL ¼ 300 K, TU ¼ 400 K. The results show that the dynamic deﬂections of the P=FGM plate are increased by increasing volume fraction index N; this is because the stiffness of the plate becomes weaker when the volume fraction index N is increased. It can also be seen that the dynamic response becomes greater when the temperature-dependent properties are taken into account.

4.4

Vibration of Postbuckled Sandwich Plates with FGM Face Sheets in Thermal Environments

Finally, we consider the small- and large-amplitude vibrations of compressively and thermally postbuckled sandwich plates with FGM face sheets in thermal environments. The length, width, and total thickness of the sandwich plate are a, b, and h. The thickness of each FGM face sheet is hF, while the thickness of the homogeneous substrate is hH (see Figure 3.14). Note that in this section the Z is in the direction of the downward normal to the middle surface. The FGM face sheet is made from a mixture of ceramics and metals, the mixing ratio of which is varied continuously and smoothly in the Z-direction. As in the case of Section 3.4, we assume the effective Young’s modulus Ef, thermal expansion coefﬁcient af, and thermal conductivity kf of FGM face sheets are functions of temperature, so that Ef, af, and kf are both temperature- and position-dependent. The Poisson ratio nf depends weakly on temperature change and is assumed to be a constant. Note that in this section, we assume the volume fraction Vm follows a simple power law. According to rule of mixture, we have

Z t0 N þ Ec (T) t1 t0 Z t0 N þ ac (T) af (Z, T) ¼ [am (T) ac (T)] t1 t0 Z t0 N þ kc (T) kf (Z, T) ¼ [km (T) kc (T)] t1 t0 Z t0 N þ rc rf (Z) ¼ (rm rc ) t1 t0 Ef (Z, T) ¼ [Em (T) Ec (T)]

(4:57a) (4:57b) (4:57c) (4:57d)

The temperature ﬁeld is assumed to be the same as in Section 4.2. The 1D steady-state heat transfer equation (Equation 3.55) will be solved and solutions are the same as Equations 3.56 through 3.60.

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Nonlinear Vibration of Shear Deformable FGM Plates

133

The plate is assumed to be geometrically perfect, and is subjected to a compressive edge load in the X-direction and=or thermal loading. Two cases of compressively postbuckled plates and of thermally postbuckled plates are considered. All four edges of the plate are assumed to be simply supported. Depending upon the in-plane behavior at the edges, two cases, case 1 (for the compressively buckled plate) and case 2 (for the thermally buckled plate), will be considered. The boundary conditions can be expressed by Equation 3.65 for these two cases. Introducing dimensionless quantities of Equations 2.8 and 4.6, and * (gT3 , gT4 , g T6 , gT7 ) ¼ (DTx , DTy , FTx , FTy )a2 =p2 h2 D11 * D22 * ]1=2 , lx ¼ Pb=4p2 [D11

lT ¼ a0 DT1

(4:58)

where a0 is an arbitrary reference value, deﬁned by Equation 3.39. As mentioned in Section 3.4 the stretching–bending coupling, which is given in terms of Bij* and Eij* (i, j ¼ 1, 2, 6), is still existed even for the midplane symmetric plate, when the plate is subjected to heat conduction. Hence the general von Kármán-type equations can be written in a similar form as expressed by Equations 1.33 through 1.36, just necessary to delete transverse applied load q, and the nonlinear motion equations can then be written in dimensionless form as L11 (W) L12 (Cx ) L13 (Cy ) þ g 14 L14 (F) L16 (MT ) ! €y €x @ C @ C 2 € þg þb ¼ g 14 b L(W, F) þ L17 (W) 80 @x @y 1 L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) g 24 L24 (W) ¼ g 24 b2 L(W, W) 2 € @W €x þ g10 C L31 (W) þ L32 (Cx ) L33 (Cy ) þ g14 L34 (F) L36 (ST ) ¼ g90 @x L41 (W) L42 (Cx ) þ L43 (Cy ) þ g14 L44 (F) L46 (ST ) ¼ g90 b

(4:59) (4:60) (4:61)

€ @W € y (4:62) þ g 10 C @y

where nondimensional linear operators Lij() and nonlinear operator L() are deﬁned by Equations 2.14 and 4.12, and the boundary conditions can be written in dimensionless form as x ¼ 0, p: W ¼ Cy ¼ 0

(4:63a)

Mx ¼ Px ¼ 0

(4:63b)

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134

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells ðp

1 p

b2 0

@2F dy þ 4lx b2 ¼ 0 @y2

ðp ðp g 224 b2 0 0

(for compressively buckled plate)

(4:63c)

@Cy @2F @2F @Cx þ g233 b g5 2 þ g24 g511 @x @y @y2 @x

2 @2W 1 @W 2 2@ W g24 þ g 244 b g 24 g611 @x2 @y2 2 @x þ g 224 gT1 g 5 g T2 )lT dx dy ¼ 0 (for thermally buckled plate)

(4:63d)

y ¼ 0, p:

ðp 0

ðp ðp 0 0

@2F dx ¼ 0 @x2

W ¼ Cx ¼ 0

(4:63e)

My ¼ P y ¼ 0

(4:63f)

(for compressively buckled plate)

(4:63g)

2 @Cy @2F @Cx 2@ F þ g g b þ g g b 5 24 220 522 @x @y @x2 @y2 2 2 @2W 1 2@ W 2 @W þ g b b g g24 g240 622 @x2 @y2 2 24 @y þ (gT2 g5 gT1 )lT dy dx ¼ 0 (for thermally buckled plate)

(4:63h)

We assume that the solution of Equations 4.59 through 4.62 can be expressed as ~ y, t) W(x, y, t) ¼ W*(x, y) þ W(x, ~ x (x, y, t) Cx (x, y, t) ¼ C*x (x, y) þ C ~ y (x, y, t) Cy (x, y, t) ¼ C*y (x, y) þ C ~ y, t) F(x, y, t) ¼ F*(x, y) þ F(x,

(4:64)

where W*(x, y) is an initial time-independent deﬂection due to pre- and postbuckling equilibrium states of sandwich plates subjected to uniaxial compression and=or thermal loading. C*x (x, y), C*y (x, y), and F*(x, y) are the midplane rotations and stress function corresponding to W*(x, y). ~ y, t) is an additional time-dependent displacement which is considered W(x,

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Nonlinear Vibration of Shear Deformable FGM Plates

135

to originate from the linear or nonlinear vibration of sandwich plates. ~ y (x, y, t), and F~(x, y, t) are deﬁned analogously to Cx*(x, y), ~ x (x, y, t), C C ~ y, t). Cy*(x, y), and F*(x, y), but is for W(x, Substituting Equation 4.64 into Equations 4.59 through 4.62, we obtain two sets of equations and can be solved in sequence. The ﬁrst set of equations yields the particular solution of static postbuckling or thermal postbuckling deﬂection, and the second set of equations gives the homogeneous solution of vibration characteristics on the buckled plate. As has been shown in Section 3.4, the prebuckling deﬂection caused by temperature ﬁeld (Shen 2007) should be included, when the plate is subjected to heat conduction. Solutions W*(x, y), Cx*(x, y), Cy*(x, y), and F*(x, y) may be expressed as h i h i

5 (3) (3) 3 W* ¼ « A(1) 11 sin kx sin ly þ « A13 sin kx sin 3ly þ A31 sin 3k sin ly þ O «

(4:65)

y2 x2 y2 x2 (2) b(0) þ «2 B(2) b(2) þ B(2) F* ¼ B(0) 00 00 00 00 20 cos 2kx þ B02 cos 2ly 2 2 2 2 2 2 y x (4) (4) (4) þ «4 B(4) b(4) þ B(4) 00 00 20 cos 2kx þ B02 cos 2ly þ B22 cos 2kx cos 2ly þ B40 cos 4kx 2 2 i (4) (4) 5 þ B(4) (4:66) 04 cos 4ly þ B24 cos 2kx cos 4ly þ B42 cos 4kx cos 2ly þ O(« )

h i h i (3) (3) 3 5 Cx* ¼ « C(1) 11 cos kx sin ly þ « C13 cos kx sin 3ly þ C31 cos 3kx sin ly þ O(« ) (4:67) h i h i (3) (3) 3 5 Cy* ¼ « D(1) 11 sin kx cos ly þ « D13 sin kx cos 3ly þ D31 sin 3kx cos ly þ O(« ) (4:68)

As is mentioned before, all coefﬁcients in Equations 4.65 through 4.68 are related and can be expressed in terms of A(1) 11 . ~ y (x, y, t), and F(x, ~ x (x, y, t), C ~ y, t), C ~ y, t) satisfy the nonlinear Then, W(x, motion equations:

~ x ) L13 (C ~ y ) þ g L14 (F) ~ L12 (C ~ þ W*, F ~ F* ~ ¼ g14 b2 L W ~ þ L W, L11 (W) 14 ! € € ~y ~ @C @C x € ~ þ L17 (W) þ g80 þb (4:69) @x @y

~ x ) þ g L23 (C ~ y ) g L24 (W) ~ ¼ 1 g24 b2 L W ~ þ 2W*, W ~ (4:70) ~ þ g24 L22 (C L21 (F) 24 24 2

€~ @W €~ þ g 10 C x @x ~€ €~ ~ x ) þ L43 (C ~ y ) þ g L44 (F) ~ L42 (C ~ ¼ g 90 b @ W þ g 10 C L41 (W) y 14 @y ~ x ) L33 (C ~ y ) þ g L34 (F) ~ þ L32 (C ~ ¼ g 90 L31 (W) 14

(4:71) (4:72)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Using the perturbation procedure as described in Section 4.2, we obtain asymptotic solutions, up to third order, as ~ y, t) ¼ «[w1 (t) þ g1 w € 1 (t)] sin mx sin ny W(x, þ («w1 (t))3 [g331 sin 3mx sin ny þ g313 sin mx sin 3ny] þ O(«4 ) (4:73) h i ~ y, t) ¼ « g(1,1) w1 (t) þ g4 w € (t) sin mx sin ny F(x, 1 31 þ («w1 (t))2 (g402 cos 2ny þ g420 cos 2mx) þ («w1 (t))3 h i (1,3) 4 g(3,1) 31 g331 sin 3mx sin ny þ g31 g313 sin mx sin 3ny þ O(« ) (4:74) h i ~ x (x, y, t) ¼ « g(1,1) w1 (t) þ g2 w € C (t) cos mx sin ny 1 11 þ («w1 (t))2 g12 sin 2mx þ («w1 (t))3 h i (1,3) 4 g(3,1) 11 g331 cos 3mx sin ny þ g11 g313 cos mx sin 3ny þ O(« ) (4:75) h i ~ y (x, y, t) ¼ « g(1,1) w1 (t) þ g3 w € C (t) sin mx cos ny 1 21 þ («w1 (t))2 g22 sin 2ny þ («w1 (t))3 h i (1,3) 4 g(3,1) 21 g331 sin 3mx cos ny þ g21 g313 sin mx cos 3ny þ O(« ) (4:76) (i,j)

(i,j)

(i,j)

Note that in Equations 4.73 through 4.76 coefﬁcients g11 , g21 , g31 (i, j ¼ 1, 3), etc. are given in detail in Appendix K. Also we have € 1 (t)] sin mx sin ny þ («w1 (t))2 (g441 cos 2mx þ g442 cos 2ny) «[g41 w1 (t) þ g43 w

þ g14 b2 («w1 (t))2 «A(1) 4k2 n2 g402 cos 2ny þ 4l2 m2 g420 cos 2mx sin kx sin ly 11

4k2 n2 g402 cos 2ny þ 36l2 m2 g420 cos 2mx sin kx sin 3ly þ g14 b2 («w1 (t))2 «3 A(3) 13

36k2 n2 g402 cos 2ny þ 4l2 m2 g420 cos 2mx sin 3kx sin ly þ g14 b2 («w1 (t))2 «3 A(3) 31 þ g42 («w1 (t))3 sin mx sin ny ¼ 0

(4:77)

Multiplying Equation 4.77 by (sin mx sin ny) and integrating over the plate area, we obtain g43

d2 («w1 ) þ g41 («w1 ) þ g44 («w1 )2 þ g42 («w1 )3 ¼ 0 dt 2

(4:78)

From Equation 4.78, the nonlinear frequency of the plates can be expressed as

vNL

9g42 g41 10g244 2 ¼ vL 1 þ A 12g241

1=2 (Px 6¼ Pcr or DT 6¼ DTcr )

(4:79a)

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Nonlinear Vibration of Shear Deformable FGM Plates

vNL ¼

3 g42 1=2 A 4 g43

137

(Px ¼ Pcr or DT ¼ DTcr )

(4:79b)

From Equation 4.79b, it can be seen that the nonlinear frequency varies linearly with the amplitude W max=h, when Px=Pcr (or DT=DTcr) ¼ 1. We ﬁrst examine the relationship between natural frequency ratio (v=v0) and in-plane compressive load ratio P=Pcr for an isotropic rectangular plate with aspect ratio of 3, where v0 is the lowest linear natural frequency of the plate and Pcr is the buckling load of the same plate under the uniaxial compression. The curves are plotted in Figure 4.8 and compared with the FEM result of Yang and Han (1983) based on high-order triangular membrane ﬁnite element combined with a fully conforming triangular plate bending element. We then examine the relationship between fundamental frequencies vL (in Hz) and temperature rise T (in K) for a simply supported Si3N4=SUS304 plate under the two special cases of isotropy, i.e., volume fraction index N ¼ 0 and 2000. The top surface is ceramic-rich, whereas the bottom surface is metal-rich. The material properties are the same as listed in Tables 1.1 through 1.3. The FGM plate has a ¼ b ¼ 0.3 m and a=h ¼ 100. The curves are plotted in Figure 4.9 and compared with the FEM result of Park and Kim (2006) based on the FSDPT.

4

3

Yang and Han (1983) Present 1: (m, n) = (5, 1)

1

2: (m, n) = (3, 1)

w /w 0

3: (m, n) = (2, 1) 2 2 1

0 0.0

3

0.5

1.0

1.5

2.0

Px/Pcr FIGURE 4.8 Comparisons of natural frequencies for the compressively postbuckled isotropic rectangular plate with aspect ratio of 3 (n ¼ 0.3).

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 800 Park and Kim (2006) Present

w L (Hz)

600

I: N = 0 II: N = 2000

400 II

200 I 0 300

310

320

330

340

350

T (K) FIGURE 4.9 Comparisons of natural frequencies for the thermally buckled Si3N4=SUS304 square plate.

These two comparisons show that the present results agree well with existing results for both compressively and thermally buckled plates. In this section, the material mixture for FGM face sheets is considered to be silicon nitride and stainless steel, referred to as Si3N4=SUS304. The material properties are the same as those adopted in Section 3.4. The plate geometric parameter a=b ¼ 1, b=h ¼ 20, and the thickness of the FGM face sheets hF ¼ 1 mm, whereas the thickness of the homogeneous substrate is taken to be hH ¼ 4, 6, and 8 mm, so that the substrate-to-face sheet thickness ratio hH=hF ¼ 4, 6, 8, respectively. Figure 4.10 shows the effects of volume fraction index N on the linear fundamental frequencies vL of the pre- and postbuckled sandwich plate with hH=hF ¼ 4 under uniform or nonuniform temperature ﬁeld. It can be seen that as the volume fraction index increases, the fundamental frequency increases in the prebuckling region, but decreases in the initial postbuckling region (Px < 3500 kN), and in the deep postbuckling region the fundamental frequency becomes greater, when increasing in N. This is due to the fact that the plate stiffness is increased in the prebuckling region, when increasing in N, but in the initial postbuckling region the initial deﬂection is an important issue and the plate will have a small deﬂection when it has a great stiffness, further in the deep postbuckling region the effect of plate stiffness becomes more pronounced again. Figure 4.11 shows temperature changes on the fundamental frequencies of the pre- and postbuckled sandwich plate with hH=hF ¼ 4 and N ¼ 2 under uniform or nonuniform temperature ﬁeld. It can be seen that, for uniform temperature ﬁeld, as the temperature is increased, fundamental frequencies

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139

12,000 TU = TL = 300 K, hH /hF = 4.0 10,000

1: N = 2000 2: N = 2.0 3: N = 0.5

w L (Hz)

8000

4: N = 0 6000 4000 1

2000 0

0

2

3 4 1000

2000

3000

4000

5000

6000

4000

5000

6000

Px (kN)

(a)

12,000 TU = 500 K, TL = 400 K hH /hF = 4.0

10,000

1: N = 2000 w L (Hz)

8000

2: N = 2.0 3: N = 0.5 4: N = 0

6000

4 3 2 1

4000 2000 0 (b)

0

1000

2000

3000 Px (kN)

FIGURE 4.10 Effects of volume fraction index N on the fundamental frequencies of the pre- and postbuckled sandwich plate in thermal environments: (a) uniform temperature ﬁeld; (b) heat conduction.

have decreased in the prebuckling region, but increased in the postbuckling region. It can also be seen that the effect of nonuniform temperature ﬁeld is larger than that of uniform temperature ﬁeld. When heat conduction is put into consideration, the larger top-bottom temperature difference leads to larger fundamental frequency rise.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 10,000

w L (Hz)

8000

6000

4000

2

1

hH/hF = 4.0, N = 2.0

2000

1: TU = TL = 300 K 2: TU = TL = 600 K

0 0

1000

2000

(a)

3000

4000

5000

Px (kN)

10000

8000

w L (Hz)

6000

4000

hH/hF = 4.0, N = 2.0

1 2 3

2000

1: TU = 600 K, TL = 400 K 2: TU = 500 K, TL = 400 K 3: TU = 400 K, TL = 400 K

0 (b)

0

1000

2000

3000

4000

5000

Px (kN)

FIGURE 4.11 Effects of temperature changes on the fundamental frequencies of the pre- and postbuckled sandwich plate: (a) uniform temperature ﬁeld; (b) heat conduction.

From Figures 4.10 and 4.11, it can be seen that, as the compressive load reaches the buckling load, the fundamental frequencies will drop to zero under uniform temperature ﬁeld. In contrast, the fundamental frequencies do not go to zero, because no bifurcation-type buckling could occur when heat conduction is taken into account. Figures 4.12 and 4.13 show, respectively, the effects of volume fraction index N, and temperature changes on the nonlinear frequency ratio of

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Nonlinear Vibration of Shear Deformable FGM Plates 3.0 1: N = 2.0 2.5

Wmax/h

2.0

3

2: N = 0.5

1 2

141

1 2 3

1 2 3

3: N = 0 I

II

III

1.5 1.0

TU = TL = 300 K, hH/hF = 4.0 I: Px/Pcr = 2 II: Px/Pcr = 1 III: Px/Pcr = 0

0.5 0.0 0.0

0.5

1.0

1.5

2.0

w NL/ω 0 FIGURE 4.12 Effects of volume fraction index N on the nonlinear frequency ratio of the pre- and postbuckled sandwich plate.

3.0 2.5

hH/hF = 4.0, N = 2.0 I II

Wmax/h

2.0

III

I: Px/P0 = 2 II: Px/P0 = 1 III: Px/P0 = 0

1.5

TU = 300 K,

1.0

TL = 300 K 0.5

TU = 600 K, TL = 300 K

0.0 0.0

0.5

1.0 w NL/w 0

1.5

2.0

FIGURE 4.13 Effects of temperature changes on the nonlinear frequency ratio of the pre- and postbuckled sandwich plate.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 3000 2500

hH/hF = 4.0 1: N = 2000 1

2: N = 2.0

2

3: N = 0.5

3

4: N = 0

w L (Hz)

2000

4

1500 1000 500 0 300

350

400

450

T (K)

500

550

600

650

FIGURE 4.14 Effects of volume fraction index N on the fundamental frequencies of the thermally pre- and postbuckled sandwich plate.

the pre- and postbuckled sandwich plate in thermal environments. Three cases, i.e., Px=Pcr ¼ 0, 1, and 2, are considered. Px=Pcr ¼ 0 denotes no in-plane loads, Px=Pcr ¼ 1 denotes bifurcation buckling case, and Px=Pcr ¼ 2 represents a large-amplitude free vibration about a postbuckled equilibrium state. Note that in Figure 4.13 Px=Pcr is replaced by Px=P0, where P0 is a reference value of buckling load of the plate at TU ¼ TL ¼ 300 K. It can be seen that the nonlinear frequency ratio is decreased, when the volume fraction index N is increased. The temperature changes only have small effects on the nonlinear frequency ratio of the plate. Note that, in the present study, the solution is based on the assumption that the vibration of the plate is symmetric about the ﬂat position. Another type of motion is possible in which the plate vibrates about a static buckled position on one side of the ﬂat position. Such motion is however not considered in the present study. Figures 4.14 and 4.15 are thermally postbuckled vibration results for the same plate analogous to the compressively postbuckled vibration results of Figures 4.10 and 4.12, which are for the thermal loading case of uniform temperature ﬁeld. They lead to broadly the same conclusions as do Figures 4.10 and 4.12.

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143

3.0 hH/hF = 4.0 2.5

I: T/Tcr = 2 II: T/Tcr = 1

Wmax/h

2.0

I

III: T/Tcr = 0

II

1.5

III

1.0

N = 2.0 N = 0.5

0.5 0.0 0.0

N=0

0.5

1.0

1.5

2.0

2.5

3.0

w NL/w 0 FIGURE 4.15 Effects of volume fraction index N on the nonlinear frequency ratio of the thermally pre- and postbuckled sandwich plate.

References Abrate S. (2006), Free vibration, buckling, and static deﬂections of functionally graded plates, Composites Science and Technology, 66, 2383–2394. Allahverdizadeh A., Naei M.H., and Bahrami M.N. (2008a), Nonlinear free and forced vibration analysis of thin circular functionally graded plates, Journal of Sound and Vibration, 310, 966–984. Allahverdizadeh A., Naei M.H., and Bahrami M.N. (2008b), Vibration amplitude and thermal effects on the nonlinear behavior of thin circular functionally graded plates, International Journal of Mechanical Sciences, 50, 445–454. Allahverdizadeh A., Rastgo A., and Naei M.H. (2008c), Nonlinear analysis of a thin circular functionally graded plate and large deﬂection effects on the forces and moments, Journal of Engineering Materials and Technology-ASME, 130, Article Number: 011009. Bhimaraddi A. and Chandrashekhara K. (1993), Nonlinear vibrations of heated antisymmetric angle-ply laminated plates, International Journal of Solids and Structures, 30, 1255–1268. Chen C.-S. (2005), Nonlinear vibration of a shear deformable functionally graded plate, Composite Structures, 68, 295–302. Chen C.-S. and Tan A.-H. (2007), Imperfection sensitivity in the nonlinear vibration of initially stresses functionally graded plates, Composite Structures, 78, 529–536. Chen C.-S., Chen T.J., and Chien R.-D. (2006), Nonlinear vibration of initially stressed functionally graded plates, Thin-Walled Structures, 44, 844–851.

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Cheng Z.-Q. and Reddy J.N. (2003), Frequency of functionally graded plates with three-dimensional asymptotic approach, Journal of Engineering Mechanics ASCE, 129, 896–900. Efraim E. and Eisenberger M. (2007), Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration, 299, 720–738. Elishakoff I., Gentilini C., and Viola E. (2005), Forced vibrations of functionally graded plates in the three-dimensional setting, AIAA Journal, 43, 2000–2007. Fung C.-P. and Chen C.-S. (2006), Imperfection sensitivity in the nonlinear vibration of functionally graded plates, European Journal of Mechanics—A=Solids, 25, 425–436. He X.Q., Ng T.Y., Sivashanker S., and Liew K.M. (2001), Active control of FGM plates with integrated piezoelectric sensors and actuators, International Journal of Solids and Structures, 38, 1641–1655. Huang X.-L. and Shen H.-S. (2004), Nonlinear vibration and dynamic response of functionally graded plates in thermal environments, International Journal of Solids and Structures, 41, 2403–2427. Huang X.-L. and Shen H.-S. (2006), Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments, Journal of Sound and Vibration, 289, 25–53. Kim Y.-W. (2005), Temperature dependent vibration analysis of functionally graded rectangular plates, Journal of Sound and Vibration, 284, 531–549. Kitipornchai S., Yang J., and Liew K.M. (2004), Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections, International Journal of Solids and Structures, 41, 2235–2257. Li Q., Iu V.P., and Kou K.P. (2008), Three-dimensional vibration analysis of functionally graded material sandwich plates, Journal of Sound and Vibration, 311, 498–515. Park J.S. and Kim J.-H. (2006), Thermal postbuckling and vibration analyses of functionally graded plates, Journal of Sound and Vibration, 289, 77–93. Pearson C.E. (1986), Numerical Methods in Engineering and Science, Van Nostrand Reinhold Company, Inc., New York, NY. Prakash T. and Ganapathi M. (2006), Asymmetric ﬂexural vibration and thermoelastic stability of FGM circular plates using ﬁnite element method, Composites Part B, 37, 642–649. Praveen G.N. and Reddy J.N. (1998), Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates, International Journal of Solids and Structures, 35, 4457–4476. Reddy J.N. (2000), Analysis of functionally graded plates, International Journal for Numerical Methods in Engineering, 47, 663–684. Shen H.-S. (2007), Nonlinear thermal bending response of FGM plates due to heat conduction, Composites Part B, 38, 201–215. Sundararajan N., Prakash T., and Ganapathi M. (2005), Nonlinear free ﬂexural vibrations of functionally graded rectangular and skew plates under thermal environments, Finite Element in Analysis and Design, 42, 152–168. Vel S.S. and Batra R.C. (2004), Three dimensional exact solution for the vibration of functionally graded rectangular plates, Journal of Sound and Vibration, 272, 703–730. Wu L., Wang H., and Wang D. (2007), Dynamic stability analysis of FGM plates by the moving least squares differential quadrature method, Composite Structures, 77, 383–394.

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Xia X.-K. and Shen H.-S. (2008), Vibration of post-buckled sandwich plates with FGM face sheets in a thermal environment, Journal of Sound and Vibration, 314, 254–274. Yang T.Y. and Han A.D. (1983), Buckled plate vibrations and large amplitude vibrations using high-order triangular elements, AIAA Journal, 21, 758–766. Yang J. and Huang, X.-L. (2007), Nonlinear transient response of functionally graded plates with general imperfections in thermal environments, Computer Methods in Applied Mechanics and Engineering, 196, 2619–2630. Yang J. and Shen H.-S. (2001), Dynamic response of initially stressed functionally graded rectangular thin plates, Composite Structures, 54, 497–508. Yang J. and Shen H.-S. (2002), Vibration characteristics and transient response of shear deformable functionally graded plates in thermal environment, Journal of Sound and Vibration, 255, 579–602. Yang J., Kitipornchai S. and Liew K.M. (2003), Large amplitude vibration of thermoelectric-mechanically stressed FGM laminated plates, Computational Methods in Applied Mechanics and Engineering, 192, 3861–3885.

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5 Postbuckling of Shear Deformable FGM Shells

5.1

Introduction

Buckling of circular cylindrical shells has posed bafﬂing problems to engineering for many years. This is due to the fact that large discrepancies between theoretical prediction and experimental results had been the focus of long debate in the case of compressive buckling of cylindrical shells. As von Kármán and Tsien (1941) argued at the time, geometrical nonlinearity must play an important part in the phenomenon of thin shell buckling. Donnell and Wan (1950) reported that the initial geometric imperfection has a signiﬁcant effect on the buckling and postbuckling behavior of cylindrical shells subjected to axial compression. In their analysis, however, the membrane prebuckling state was assumed, like von Kármán and Tsien (1941) did and, therefore, the boundary conditions cannot be incorporated accurately. The importance of the nonlinear prebuckling deformations and its role in the buckling analysis of cylindrical shells has been discussed by Stein (1962, 1964). On the other hand, Koiter (1945) provided a general theory of the initial postbuckling behavior of elastic bodies under static conservative load. Following the pioneer works of Kármán and Tsien (1941), Donnell and Wan (1950), Stein (1962, 1964), and Koiter (1945, 1963), numerous researches have been made on this topic. The problem may be considered to be solved completely in the domain of homogeneous, isotropic elastic materials. In the design of an FGM shell as well as a homogeneous, isotropic shell, it is of technical importance to examine its resistance to buckling under expected loading conditions. For that purpose, the determination of the buckling load alone is not sufﬁcient in general, but it is further required to clarify the postbuckling behavior, that is, the behavior of the shell after passing through the buckling load. One of the reasons is to estimate the effect of practically unavoidable imperfections on the buckling load and the second is to evaluate the ultimate strength to exploit the load-carrying capacity of the shell structure. Shen (2002, 2003) provided, respectively, the postbuckling solutions of FGM cylindrical shells under axial compression and external pressure in thermal environments. In his studies, a boundary layer theory for the shell 147

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buckling suggested by Shen and Chen (1988, 1990) was adopted. However, because the shells were considered as being relatively thin and therefore the transverse shear deformation was not accounted for. These works were then extended to the case of FGM hybrid cylindrical shells under axial compression, external pressure, or their combination in thermal environments by Shen (2005), and Shen and Noda (2005, 2007) based on a higher order shear deformation shell theory. Furthermore, Shahsiah and Eslami (2003a,b) performed the thermal buckling of FGM cylindrical shells under three types of thermal loading as uniform temperature rise, linear and nonlinear temperature variation through the thickness, based on the ﬁrst-order shear deformation shell theory. Similar work was then done by Wu et al. (2005) based on classical shell theory of Donnell-type. Subsequently, the effect of initial geometric imperfections and applied actuator voltage on thermal buckling of FGM cylindrical shells was discussed by Mirzavand et al. (2005) and Mirzavand and Eslami (2006, 2007). As we all know, the imperfect cylindrical shell only has limit point load, which could be obtained by solving nonlinear governing equations as initial postbuckling or full postbuckling analysis. Based on a higher order shear deformation shell theory, Woo et al. (2005) presented Fourier series solutions for the thermomechanical postbuckling of FGM plates and shallow shells, from which the results for an initially heated cylindrical shell were obtained as a limiting case. Soﬁyev (2007) studied linear free vibration and buckling of FGM laminated cylindrical shells by using Galerkin method. Sheng and Wang (2008) performed the linear vibration, buckling and dynamic stability of FGM cylindrical shells embedded in an elastic medium and subjected to mechanical and thermal loads based on the ﬁrst-order shear deformation shell theory. In the above studies, however, the materials properties were virtually assumed to be temperature-independent (T-ID). Moreover, Shen (2004, 2007) provided a thermal postbuckling analysis for FGM cylindrical shells under uniform temperature ﬁeld or heat conduction based on classical shell theory and higher order shear deformation shell theory, respectively. In his study, the material properties were considered to be temperature-dependent (T-D) and the effect of imperfections on the thermal postbuckling response was reported. Recently, Kadoli and Ganesan (2006) presented linear thermal buckling and free vibration analysis for FGM cylindrical shells with clamped boundary conditions. In their analysis, the material properties were assumed to be temperature-dependent, and ﬁnite element equations based on the ﬁrst-order shear deformation shell theory were formulated. On the other hand, due to the temperature gradient the shell is subjected to additional moments along with the membrane forces and the problem cannot be posed as an eigenvalue problem, i.e., no buckling temperature is evident, when the edges of the shell are simply supported. Therefore, the existing solutions for simply supported FGM shells subjected to transverse temperature variation, i.e., linear and=or nonlinear gradient through the thickness, may be physically incorrect.

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Postbuckling of Shear Deformable FGM Shells

5.2

149

Boundary Layer Theory for the Buckling of FGM Cylindrical Shells

It has been shown in Shen and Chen (1988, 1990) that in shell buckling, there exists a boundary layer phenomenon where prebuckling and buckling displacements vary rapidly. Consider a circular cylindrical shell with mean radius R, length L, and thickness h. The shell is referred to a coordinate system (X, Y, Z) in which X and Y are in the axial and circumferential directions of the shell and Z is in the direction of the inward normal to the middle surface (see Figure 5.1). The corresponding displacements are designated by U, V, and W. Cx and Cy are the rotations of normals to the middle surface with respect to the Y- and X-axes, respectively. The origin of the coordinate system is located at the end of the shell on the middle plane. The shell is assumed to be relatively thick, geometrically imperfect. Denoting the initial geometric imperfection by W*(X, Y), let W(X, Y) be the additional deﬂection and F(X, Y) be the stress function for the stress resultants deﬁned by N x ¼ F,yy, N y ¼ F,xx, and N xy ¼ F,xy, where a comma denotes partial differentiation with respect to the corresponding coordinates. 5.2.1

Donnell Theory

In 1933, Donnell established his nonlinear theory of circular cylindrical shells, in connection with the analysis of torsional buckling of thin-walled tubes. Owing to its relative simplicity and practical accuracy, this theory has been the most widely used for analyzing both buckling and postbuckling problems, despite criticism concerning its applicability. h

L X Z Y

2R

FIGURE 5.1 Geometry and coordinate system of a cylindrical shell.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

The Donnell theory is based on the following assumptions (see Yamaki 1984): 1. The shell is sufﬁciently thin, i.e., h=R 1 and h=L 1. 2. The strains « are sufﬁciently small, i.e., « 1, and, therefore, Hooke’s law holds. 3. Straight lines normal to the undeformed middle surface remain straight and normal to the deformed middle surface with their length unchanged. 4. The normal stress acting in the direction normal to the middle surface may be neglected in comparison with the stresses acting in the direction parallel to the middle surface. 5. In-plane displacements U and V are inﬁnitesimal, while normal displacement W is of the same order as the shell thickness, i.e., jUj h, jVj h, and jWj ¼ O(h). 6. The derivatives of W are small, but their squares and products are of the same order as the strain « considered. As a result " 2 2 # @W @W @W @W @W @W , , ¼ O(«): @X , @Y 1, @X @X @Y @Y 7. Curvature changes are small and the inﬂuence of U and V are negligible so that they can be represented by linear function of W only. The assumptions (3) and (4) constitute the so-called Kirchhoff–Love hypotheses while those from (5) to (7) correspond to the shallow shell approximations applicable for deformations dominated by the normal displacement W. 5.2.2

Generalized Kármán–Donnell-Type Nonlinear Equations

The classical nonlinear shell theory suggested by Donnell (1933) does not include the transverse shear and normal stresses and strains, which will play an important role in the analysis of composite shell structures. Reddy and Liu (1985) developed a simple higher order shear deformation shell theory, in which the transverse shear strains are assumed to be parabolically distributed across the shell thickness and which contains the same number of dependent unknowns as in the ﬁrst-order shear deformation theory, and no shear correction factors are required. The governing equations of motion were derived by means of the principle of virtual work and can be expressed as

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151

@N 1 @N 6 @2U @ 2 Cx @3W þ ¼ I1 2 þ I 2 c1 I4 2 @X @Y @t @t @X@t2 @ 2 Cy @N 6 @N 2 @2V @3W c 1 I4 þ ¼ I1 2 þ I 2 2 @X @Y @t @t @Y@t2 @Q1 @Q2 @ @W @W @ @W @W N2 N1 N6 þ þ þ N6 þ þ N2 @X @Y @X R @X @Y @Y @X @Y 2 2 2 @R1 @R2 @ P1 @ P6 @ P2 þ2 þ þ c1 þ þ q c2 @X @Y @X2 @X@Y @Y2 ! @2W 2 @2 @2W @2W @ 2 @U @V @ 2 @Cx @Cy þ þ ¼ I1 2 c1 I7 2 þ c 1 I4 2 þ þ c1 I 5 2 @t @X2 @Y2 @t @X @Y @t @X @Y @t @M1 @M6 @P1 @P6 @2U @ 2 Cx @3W þ Q1 þ C2 R1 C1 þ ¼ I2 2 þ I3 c I 1 5 @X @Y @X @Y @t2 @t @X@t2 @ 2 Cy @M6 @M2 @P6 @P2 @2V @3W c1 I 5 (5:1) þ Q2 þ c2 R2 c1 þ ¼ I2 2 þ I3 2 @X @Y @X @Y @t @t @Y@t2

All symbols used in Equation 5.1 are deﬁned as in Equation 1.26. * , B*ij , D*ij , E*ij , F*ij , Introducing the reduced stiffness matrices A ij and H*ij (i, j ¼ 1, 2, 6) and by using the same manner described in Section 1.4, we derive the generalized Kármán–Donnell-type nonlinear equations of motion, which can be expressed in terms of a stress function F, two rotations Cx and Cy, and a transverse displacement W, along with the initial geometric imperfection W*. These equations are then extended to the case of shear deformable FGM cylindrical shells including thermal effects. ~12 (Cx ) L ~13 (Cy ) þ L ~14 (F) L ~15 (N T ) L ~16 (MT ) 1 F,xx ~11 (W) L L R € € € ~ ~ ¼ L(W þ W*, F) þ L (W) I (C , þ C , ) þ q 17

8

x x

y y

(5:2)

~21 (F) þ L ~22 (Cx ) þ L ~23 (Cy ) L ~24 (W) L ~25 (N T ) þ 1 W,xx ¼ 1 L(W ~ L þ 2W*, W) (5:3) R 2 € € I C ~31 (W) þ L ~32 (Cx ) L ~33 (Cy ) þ L ~34 (F) L ~35 (N T ) L ~36 (ST ) ¼ I 5 W, L 3 x x

(5:4)

€ € I C ~41 (W) L ~42 (Cx ) þ L ~43 (Cy ) þ L ~44 (F) L ~45 (N T ) L ~46 (ST ) ¼ I 5 W, L 3 y y

(5:5)

Note that the geometric nonlinearity in the von Kármán sense is given ~ in Equations 5.2 and 5.3, and the other linear operators in terms of L() ~ij() are deﬁned by Equation 1.37, and the forces, moments and higher L order moments caused by elevated temperature are deﬁned by Equation 1.28.

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5.2.3

Boundary Layer-Type Equations

Introducing the following dimensionless quantities x ¼ pX=L,

y ¼ Y=R,

b ¼ L=pR, Z ¼ L2 =Rh,

* D22 * A11 * A22 * ]1=4 « ¼ (p2 R=L2 )[D11 * D22 * A11 * A22 * ]1=4 , (W, W*) ¼ «(W, W*)=[D11

* D22 * ]1=2 F ¼ «2 F=[D11

* D22 * A11 * A22 * ]1=4 (Cx , Cy ) ¼ «2 (Cx , Cy )(L=p)=[D11 * =D11 * ]1=2 , g14 ¼ [D22

* =A22 * ]1=2 , g24 ¼ [A11

* =A22 * g5 ¼ A12

* , (Mx , Px ) ¼ « (Mx , 4Px =3h2 )L2=p2 D11 * [D11 * D22 * A11 * A22 * ]1=4 Nx ¼ « N k L =p D11 * A22 * =D11 * D22 * ]1=4 (gT1 , gT2 ) ¼ ATx , ATy R[A11 sﬃﬃﬃﬃﬃ pt E0 I1 E0 L2 c1 E0 (I5 I1 I4 I2 ) , g170 ¼ 2 t¼ , g171 ¼ * * p r0 D11 L r0 r0 I1 D11 2

2

2

(g80 , g90 , g10 ) ¼ (I 8 , I 5 , I 3 )

2

E0 , * r0 D11

* A22 * ]1=8 =4p[D11 * D22 * ]3=8 lq ¼ q(3)3=4 LR3=2 [A11

(5:6)

in which E0 and r0 are the reference values. The nonlinear equations (Equations 5.11 through 5.14) may then be written in dimensionless form as «2 L11 (W) «L12 (Cx ) «L13 (Cy ) þ «g 14 L14 (F) «L15 (N T ) «L16 (MT ) g14 F,xx ! € € € þ «g80 @ Cx þ b @ Cy þ g 14 4 (3)1=4 lq «3=2 ¼ g14 b2 L(W þ W*, F) þ «2 L17 (W) @x @y 3

L21 (F) þ g24 L22 (Cx ) þ g 24 L23 (Cy ) «g24 L24 (W) L25 (N T ) þ g24 W,xx 1 ¼ g 24 b2 L(W þ 2W*,W) 2 «L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) L35 (N T ) L36 (ST ) € @W €x þ g10 C ¼ «g90 @x «L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) L45 (N T ) L46 (ST ) € @W €y þ g 10 C ¼ «g90 b @y

(5:7)

(5:8)

(5:9)

(5:10)

where all the dimensionless operators Lij() and L() are deﬁned by Equations 2.14 and 4.12. In Equation 5.6, we introduce an important parameter «. For most FGMs * D22 * A11 * A22 * 1=4 ﬃ 0.3h. Furthermore, when Z ¼ (L2=Rh) > 2.96, then from ½D11 Equation 5.6 « < 1. In particular, for homogeneous isotropic cylindrical shells,

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pﬃﬃﬃﬃﬃ « ¼ p2 =ZB 12, where ZB ¼ (L2 =Rh)[1 n2 ]1=2 is the Batdorf shell parameter, which should be greater than 2.85 in the case of classical linear buckling analysis (Batdorf 1947). In practice, the shell structure will have Z > 10, so that we always have « 1. When « < 1, Equations 5.7 through 5.10 are of the boundary layer type. We will show that, in Section 5.3, (1) the nonlinear prebuckling deformations, large deﬂections in the postbuckling range and initial geometric imperfections of the shell could be considered simultaneously; (2) the full postbuckling and imperfection sensitivity analysis could be performed; and (3) it is no longer to guess the forms of solutions which will be obtained step by step, and such solutions satisfy both governing equations and boundary conditions accurately in the asymptotic sense.

5.3

Postbuckling Behavior of FGM Cylindrical Shells under Axial Compression

The buckling and postbuckling behavior of axially compressed FGM cylindrical shells is a major concern for structural instability and it represents one of the best known examples of the very complicated stability behavior of shell structures. It is assumed that the effective Young’s modulus Ef, thermal expansion coefﬁcient af, and thermal conductivity kf are assumed to be functions of temperature, so that Ef, af, and kf are both temperature- and position-dependent. The Poisson ratio nf depends weakly on temperature change and is assumed to be a constant. We assume the volume fraction Vm follows a simple power law. According to rule of mixture, we have

Z t1 N þEc (T) t2 t1 Z t1 N þac (T) af (Z, T) ¼ [am (T) ac (T)] t2 t1 Z t1 N þkc (T) kf (Z, T) ¼ [km (T) kc (T)] t2 t1 Ef (Z, T) ¼ [Em (T) Ec (T)]

(5:11a) (5:11b) (5:11c)

It is evident that when Z ¼ t1 , Ef ¼ Ec , af ¼ ac , and kf ¼ kc , and when Z ¼ t2 , Ef ¼ Em , af ¼ am , and kf ¼ km . We assume that the temperature variation occurs in the thickness direction only and one-dimensional temperature ﬁeld is assumed to be constant in the XY plane of the shell. In such a case, the temperature distribution along the thickness can be obtained by solving a steady-state heat transfer equation

d dT k ¼0 dZ dZ

(5:12)

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Equation 5.12 is solved by imposing the boundary conditions T ¼ TU at Z ¼ t1 and T ¼ TL at Z ¼ t2, and the solution, by means of polynomial series, is Tf ¼ TU þ (TL TU )h(Z)

(5:13)

in which 1 h(Z) ¼ C

"

kmc Z t1 Nþ1 k2mc Z t1 2Nþ1 þ (N þ 1)kc t2 t1 (2N þ 1)k2c t2 t1 k3mc Z t1 3Nþ1 k4mc Z t1 4Nþ1 þ (3N þ 1)k3c t2 t1 (4N þ 1)k4c t2 t1 # k5mc Z t1 5Nþ1 (5:14a) (5N þ 1)k5c t2 t1 Z t1 t2 t1

C¼1

kmc k2mc k3mc k4mc þ þ (N þ 1)kc (2N þ 1)k2c (3N þ 1)k3c (4N þ 1)k4c

k5mc (5N þ 1)k5c

(5:14b)

where kmc ¼ km kc . In the case of axial compression, since there is no transverse dynamic load applied, the nonlinear governing Equations 5.2 through 5.5 can be written in simple forms as ~12 (Cx ) L ~13 (Cy ) þ L ~14 (F) L ~15 (N T ) L ~16 (MT ) 1 F,xx ~ 11 (W) L L R ~ ¼ L(W þ W*, F) ~21 (F) þ L ~22 (Cx ) þ L ~23 (Cy ) L ~24 (W) L ~25 (N T ) þ 1 W,xx L R 1~ ¼ L(W þ 2W*, W) 2

(5:15)

(5:16)

~32 (Cx ) L ~33 (Cy ) þ L ~34 (F) L ~35 (N T ) L ~36 (ST ) ¼ 0 ~31 (W) þ L L

(5:17)

~41 (W) L ~42 (Cx ) þ L ~43 (Cy ) þ L ~44 (F) L ~45 (N T ) L ~46 (ST ) ¼ 0 L

(5:18)

The two end edges of the shell are assumed to be simply supported or clamped, so that the boundary conditions are X ¼ 0, L: W ¼ Cy ¼ 0,

Mx ¼ Px ¼ 0 (simply supported)

(5:19a)

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W ¼ Cx ¼ Cy ¼ 0 (clamped)

(5:19b)

2pR ð

N x dY þ 2pRhsx ¼ 0

(5:19c)

0

where sx is the average axial compressive stress Mx is the bending moment Px is the higher order moment as deﬁned by Equation 1.32 Also, we have the closed (or periodicity) condition 2pR ð

0

@V dY ¼ 0 @Y

(5:20a)

or 2pR ð "

0

@2F @2F * * A22 þ A 12 @X2 @Y2

@Cy 4 @Cx 4 * 2 E21 * * 2 E22 * þ B22 þ B21 @X @Y 3h 3h

2 4 @2W @2W W 1 @W @W @W* * * þ E þ E 21 22 3h2 @X2 @Y2 R 2 @Y @Y @Y # T T * N x þ A22 * N y dY ¼ 0 A12

(5:20b)

Because of Equation 5.20, the in-plane boundary condition V ¼ 0 (at X ¼ 0, L) is not needed in Equation 5.19. The average end-shortening relationship is deﬁned by x 1 ¼ L 2pRL

2pR ð ðL

0

0

@U 1 dXdY ¼ @X 2pRL

2pR ð ðL "

* A11 0

0

@2F @2F * þ A 12 @Y2 @X2

@Cy 4 @Cx 4 4 @2W @2W * 2 E11 * * 2 E12 * * * þ B12 2 E11 þ E þ B11 12 @X @Y 3h 3h 3h @X2 @Y2 # 2 1 @W @W @W* T T * N x þ A12 * N y dXdY (5:21) A11 2 @X @X @X where Dx is the shell end-shortening displacements in the X-direction. Introducing dimensionless quantities of Equation 5.6, and * D22 * =A11 * A22 * ]1=4 , lp ¼ sx =(2=Rh)[D11 * D22 * A11 * A22 * ]1=4 dp ¼ (x =L)=(2=R)[D11

(5:22)

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Also let "

ATx

#

ATy

ðt2 " T ¼ t1

Ax Ay

# T(Z)dZ

(5:23)

where T ¼ (TU þ TL 2T0 )=2. When T ¼ 0, ATx ( ¼ ATy ) can be found in Appendix B, and if DT 6¼ 0, then ATx can be found in Appendix D. The nonlinear equations (Equations 5.15 through 5.18) may then be written in dimensionless form (boundary layer type) as «2 L11 (W) «L12 (Cx ) «L13 (Cy ) þ «g 14 L14 (F) g14 F,xx ¼ g14 b2 L(W þ W*, F) (5:24)

L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) «g24 L24 (W) þ g 24 W,xx 1 ¼ g24 b2 L(W þ 2W*, W) 2

(5:25)

«L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) ¼ 0

(5:26)

«L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) ¼ 0

(5:27)

As mentioned in Section 5.2.3, all the dimensionless operators Lij() and L() are deﬁned by Equation 2.14. The boundary conditions expressed by Equation 5.19 become x ¼ 0, p: W ¼ Cy ¼ 0,

Mx ¼ Px ¼ 0 (simply supported)

W ¼ Cx ¼ Cy ¼ 0 (clamped) 1 2p

2p ð

b2 0

@2F dy þ 2 lp « ¼ 0 @y2

(5:28a) (5:28b) (5:28c)

and the closed condition becomes 2p ð

0

2 @Cy @2F @Cx 2@ F þ g522 b þ g 24 g220 g5 b @x @y @x2 @y2

2 2 @2W 1 2@ W 2 @W þ g þ g b W b g «g 24 g 240 622 24 @x2 @y2 2 24 @y @W @W* þ «(gT2 g5 g T1 )T dy ¼ 0 g 24 b2 @y @y

(5:29)

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It has been shown (Shen 2008a) that the effect of the boundary layer on the solution of a shell in compression is of the order «1, hence the unit endshortening relationship may be written in dimensionless form as 2p ð ðp " 2 1 @2F 1 2 2@ F dp ¼ 2 « g5 2 g24 b 4p g24 @y2 @x 0 0 2 @Cy @Cx @2W 2@ W þ g233 b «g 24 g 611 þ g 244 b þ g24 g511 @x2 @y2 @x @y # 2

1 @W @W @W* g 24 þ « g 224 gT1 g5 g T2 T dx dy g 24 2 @x @x @x

(5:30)

By virtue of the fact that T ½¼ ðTU þ TL 2T0 Þ=2 ¼ ðTU þ TL Þ=2 is assumed to be uniform, the thermal effects in Equations 5.15 through 5.18 vanish, but terms in DT intervene in Equations 5.29 and 5.30. From Equations 5.15 through 5.30, one can determine the postbuckling behavior of perfect and imperfect FGM cylindrical shells subjected to axial compression in thermal environments by means of a singular perturbation technique. The essence of this procedure, in the present case, is to assume that ~ j, y, «) þ W(x, ^ z, y, «) W ¼ w(x, y, «) þ W(x, ~ j, y, «) þ F(x, ^ z, y, «) F ¼ f (x, y, «) þ F(x, ~ x (x, j, y, «) þ C ^ x (x, z, y, «) Cx ¼ cx (x, y, «) þ C ~ y (x, j, y, «) þ C ^ y (x, z, y, «) Cy ¼ c (x, y, «) þ C

(5:31)

y

where « is a small perturbation parameter (provided Z > 2.96) as deﬁned in Equation 5.6 and w(x, y, «), f (x, y, «), cx (x, y, «), cy (x, y, «) are called outer solu~ x (x, , y, «), ~ , y, «), F(x, ~ , y, «), C tions or regular solutions of the shell, W(x, ^ ^ ~ ^ ^ Cy (x, j, y, «), and W(x, z, y, «), F(x, z, y, «), Cx (x, z, y, «), Cy (x, z, y, «) are the boundary layer solutions near the x ¼ 0 and x ¼ p edges, respectively, and j and z are the boundary layer variables, deﬁned by pﬃﬃﬃ pﬃﬃﬃ j ¼ x= «, z ¼ (p x)= «

(5:32)

This means for homogeneous pﬃﬃﬃﬃﬃ isotropic cylindrical shells the width of the boundary layers is of order Rt. In Equation 5.31, the regular and boundary layer solutions are expressed in the form of perturbation expansions as w(x, y, «) ¼

X

«j wj (x, y),

j¼1

cx (x, y, «) ¼

X j¼1

f (x, y, «) ¼

X j¼0

«j (cx )j (x, y),

cy (x, y, «) ¼

«j fj (x, y) X j¼1

«j (cy )j (x, y)

(5:33a)

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~ j, y, «) ¼ W(x,

X

~ jþ1 (x, j, y), F(x, ~ j, y, «) ¼ «jþ1 W

j¼0

~ x (x, j, y, «) ¼ C

X

X

~jþ2 (x, j, y) «jþ2 F

j¼0

«

jþ3=2

~ x) ~ (C jþ3=2 (x, j, y), Cy (x, j, y, «) ¼

X

j¼0

^ z, y, «) ¼ W(x,

X

j¼0

^ jþ1 (x, z, y), F(x, ^ z, y, «) ¼ «jþ1 W

j¼0

^ x (x, z, y, «) ¼ C

X

~ y ) (x, j, y) «jþ2 (C jþ2

X

(5:33b) ^jþ2 (x, z, y) «jþ2 F

j¼0

^ x) ^ «jþ3=2 (C jþ3=2 (x, z, y), Cy (x, z, y, «) ¼

X

j¼0

^ y ) (x, z, y) «jþ2 (C jþ2

j¼0

(5:33c) Substituting Equation 5.31 into Equations 5.24 through 5.27 and collecting the terms of the same order of «, we can obtain three sets of perturbation equations for the regular and boundary layer solutions, respectively, e.g., the regular solutions w(x, y), f(x, y) cx(x, y), and cy(x, y) should satisfy O(«0 ): g14 (f0 ),xx ¼ g 14 b2 L(w0 , f0 )

(5:34a)

1 L21 ( f0 ) þ g 24 L22 (cx0 ) þ g24 L23 (cy0 ) þ g 24 (w0 ),xx ¼ g 24 b2 L(w0 , w0 ) (5:34b) 2 L32 (cx0 ) L33 (cy0 ) þ g14 L34 ( f0 ) ¼ 0

(5:34c)

L42 (cx0 ) þ L43 (cy0 ) þ g14 L44 ( f0 ) ¼ 0

(5:34d)

O(«1 ): L12 (cx0 ) L13 (cy0 ) þ g 14 L14 ( f0 ) g14 ( f1 ),xx ¼ g 14 b2 ½Lðw1 , f0 Þ þ Lðw0 , f1 Þ

(5:35a)

L21 ( f1 ) þ g24 L22 (cx1 ) þ g 24 L23 (cy1 ) g24 L24 (w0 ) þ g24 (w1 ),xx 1 ¼ g 24 b2 L(w0 , w1 ) 2

(5:35b)

L31 (w0 ) þ L32 (cx1 ) L33 (cy1 ) þ g14 L34 ( f1 ) ¼ 0

(5:35c)

L41 (w0 ) L42 (cx1 ) þ L43 (cy1 ) þ g14 L44 ( f1 ) ¼ 0

(5:35d)

O(«2 ): L11 (w0 ) L12 (cx1 ) L13 (cy1 ) þ g14 L12 ( f1 ) g 14 ( f2 ),xx ¼ g 14 b2 ½Lðw2 þ W*, f0 Þ þ L(w1 , f1 ) þ L(w0 , f2 )

(5:36a)

L21 ( f2 ) þ g24 L22 (cx2 ) þ g 24 L23 (cy2 ) g24 L24 (w1 ) þ g24 (w2 ),xx 1 ¼ g24 b2 ½Lðw1 , w1 Þ þ Lðw2 þ 2W*, w0 Þ 2

(5:36b)

L31 (w1 ) þ L32 (cx2 ) L33 (cy2 ) þ g14 L34 ( f2 ) ¼ 0

(5:36c)

L41 (w1 ) L42 (cx2 ) þ L43 (cy2 ) þ g14 L44 ( f2 ) ¼ 0

(5:36c)

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Because of the deﬁnition of W given in Equation 5.6, we assume that w0 ¼ 0 (0) 2 and w1 ¼ A(1) 00 , along with cx0 ¼ cy0 ¼ cx1 ¼ cy1 ¼ 0, f0 ¼ B00 y =2, and (1) 2 f1 ¼ B00 y =2. The initial buckling mode is assumed to have the form (2) w2 (x, y) ¼ A(2) 11 sin mx sin ny þ A02 cos 2ny

(5:37)

and the initial geometric imperfection is assumed to have the form * sin mx sin ny ¼ «2 mA(2) W*(x, y, «) ¼ «2 a11 11 sin mx sin ny

(5:38)

* =A(2) where m ¼ a11 11 is the imperfection parameter. From Equation 5.36 one has y2 þ B(2) 11 sin mx sin ny 2 ¼ C(2) 11 cos mx sin ny

f2 ¼ B(2) 00 cx2

cy2 ¼

D(2) 11

(5:39)

sin mx cos ny

and B(2) 11 ¼

g24 m2 (2) A11 , g06

C(2) 11 ¼ g 14 m b2 B(0) 00

g01 (2) B , g00 11

D(2) 11 ¼ g 14 nb

g02 (2) B , g00 11

g24 m2 ¼ (1 þ m)g06

(5:40)

where gij are given in detail in Appendix L. ~ j, y) and Now we turn our attention to the boundary layer solutions W(x, ~ F(x, j, y) which should satisfy boundary layer equations of each order, for example O(«2 ): g 110

~ x(3=2) ~1 ~2 ~2 @3C @4W @4F @2F g þ g g g ¼0 120 14 140 14 4 3 4 @j @j @j @j2

~ x(3=2) ~1 ~1 ~2 @3C @4F @4W @2W þ g 24 g220 g24 g 240 þ g24 ¼0 4 3 4 @j @j @j @j2 g310

~ x(3=2) ~1 ~2 @2C @3W @3F g þ g g ¼0 320 14 220 3 2 @j @j @j3 g 430

~ y2 @2C @j2

¼0

(5:41a) (5:41b) (5:41c) (5:41d)

which leads to ~1 ~1 @4W @2W ~1 ¼ 0 þ 2c þ b2 W 4 @j @j2

(5:42)

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where c¼

g14 g 24 g320 g15 , 2g16

b¼

1=2 g 14 g24 g 2320 g16

(5:43)

where g15 and g16 are also given in detail in Appendix L. The solution of Equation 5.42 can be written as ~ 1 ¼ a(1) sin fj þ a(1) cos fj eqj (5:44a) W 10 01 and ~2 ¼ b(2) sin fj þ b(2) cos fj eqj F 10 01 ~ x(3=2) ¼ c(3=2) sin fj þ c(3=2) cos fj eqj C 10 01 ~ y2 ¼ 0 C

(5:44b) (5:44c) (5:44d)

where

bc q¼ 2

1=2 ,

b þ c 1=2 f¼ 2

(5:45)

^ ^ Similarly, the boundary layer solutions W(x, z, y) and F(x, z, y) can be obtained in the same manner. Then matching the regular solutions with the boundary layer solutions at ~ 1 ) ¼ (w1 þ W ~ 1 ),x ¼ 0 and the each end of the shell, e.g., let (w1 þ W ^ ^ (w1 þ W1 ) ¼ (w1 þ W1 ),x ¼ 0 at x ¼ 0 and x ¼ p edges, respectively, so that the asymptotic solutions satisfying the clamped boundary conditions are constructed as

x x x (1) (1) (1) p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ þ a exp q W ¼ « A(1) A a cos f sin f 00 00 01 10 « « « p x p x p x (1) (1) (1) A00 a01 cos f pﬃﬃﬃ þ a10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « « h (2) (2) (2) þ «2 A11 sin mx sin ny þ A02 cos 2ny A02 cos 2ny x x x (1) p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ cos f sin f þ a exp q a(1) 01 10 « « « p x px px (1) A(2) a(1) exp q pﬃﬃﬃ 02 cos 2ny 01 cos f pﬃﬃﬃ þ a10 sin f pﬃﬃﬃ « « « h i h (3) (4) (4) 4 þ «3 A(3) 11 sin mx sin ny þ A02 cos 2ny þ « A00 þ A11 sin mx sin ny i (4) (4) (4) 5 þ A(4) 20 cos 2mx þ A02 cos 2ny þ A13 sin mx sin 3ny þ A04 cos 4ny þ O(« )

(5:46)

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y2 y2 y2 2 þ « B(1) þ « þ B(2) B(2) 00 00 11 sin mx sin ny 2 2 2 x x x (2) (2) p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ þ A(1) þ b exp q b cos f sin f 00 01 10 « « « px px px (1) (2) (2) þ A00 b01 cos f pﬃﬃﬃ þ b10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « «

2 y x x x (3) (2) (3) (3) p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ þ «3 B(3) þ B þ b exp q cos 2ny þ A cos 2ny b cos f sin f 00 02 02 01 10 « « « 2 px px px (2) (3) (3) þ A02 cos 2ny b01 cos f pﬃﬃﬃ þ b10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « «

2 y (4) (4) 5 þ «4 B(4) þ B(4) (5:47) 00 20 cos 2mx þ B02 cos 2ny þ B13 sin mx sin 3ny þ O(« ) 2

F ¼ B(0) 00

x x px px (3=2) (1) (3=2) p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ p ﬃﬃ ﬃ exp q þ A exp q Cx ¼ «3=2 A(1) c sin f c sin f 00 10 00 10 « « « «

h i x x (5=2) (2) (2) þ «2 C11 cos mx sin ny þ «5=2 A02 cos 2ny c10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « h i p x p x (5=2) þ A(2) þ «3 C(3) 02 cos 2ny c10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ 11 cos mx sin ny « « h i (4) (4) (4) 4 (5:48) þ « C11 cos mx sin ny þ C20 sin mx þ C13 cos mx sin 3ny þ O(«5 ) h i h i (3) (3) 3 Cy ¼ «2 D(2) 11 sin mx cos ny þ « D11 sin mx cos ny þ D02 sin 2ny h i (4) (4) 5 þ «4 D(4) 11 sin mx cos ny þ D02 sin 2ny þ D13 sin mx cos 3ny þ O(« )

(5:49)

Note that all of the coefﬁcients in Equations 5.46 through 5.49 are related (2) and can be expressed in terms of A(2) 11 , as shown in Equation 5.40 for B11 , (j) except for A00 (j ¼ 1–4) which may be determined by substituting Equations 5.46 through 5.49 into closed condition (Equation 5.29), and then we obtain A(1) 00 ¼ 2

g5 1 lp [g g5 g T1 ]DT g 24 g 24 T2

(3) A(2) 00 ¼ A00 ¼ 0 2 2 1 2 2 (2) (2) 2 2 ¼ b (1 þ 2m) A þn b A A(4) n 00 11 02 8

(5:50)

Next, upon substitution of Equations 5.46 through 5.49 into the boundary condition (Equation 5.28c) and into Equation 5.30, the postbuckling equilibrium paths can be written as 2 4 (2) (2) (2) (4) lp ¼ l(0) p lp A11 « þlp A11 « þ

(5:51)

2 4 (T) (2) (4) A(2) A(2) dp ¼ d(0) x dx þ d x 11 « þdx 11 « þ

(5:52)

and

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

In Equations 5.51 and 5.52, A(2) « is taken as the second perturbation 11 parameter relating to the dimensionless maximum deﬂection. From Equation 5.46, by taking (x, y) ¼ (p=2m, p=2n), one has 2 A(2) 11 « ¼ Wm 2 Wm þ

(5:53)

where Wm is the dimensionless form of maximum deﬂection of the shell that can be written as " # 1 h W þ 1 Wm ¼ C3 ½D11 * D22 * A11 * A22 * 1=4 h

(5:54)

All symbols used in Equations 5.51 through 5.54 are also described in detail in Appendix L. The perturbation scheme described here is quite different from that of traditional one (Koiter 1963, Dym and Hoff 1968). In the present analysis, « is deﬁnitely a small perturbation parameter, but in the second step (A(2) 11 «) may be large. In contrast, the perturbation parameter « is deﬁned as a characteristic amplitude nondimensionalized with respect to the shell thickness (Dym and Hoff 1968), which is no longer a small perturbation parameter in the deep postbuckling range when the shell deﬂection is sufﬁciently large, e.g., W=h > 1, and in such a case the solution is questionable. Equations 5.51 through 5.54 can be employed to obtain numerical results for full nonlinear postbuckling load–shortening or load–deﬂection curves of FGM cylindrical shells subjected to axial compression. The initial buckling load of a perfect shell can readily be obtained numerically, by setting W*=h ¼ 0 (or m ¼ 0), while taking W=h ¼ 0 (note that Wm 6¼ 0). In this case, the minimum buckling load is determined by considering Equation 5.51 for various values of the buckling mode (m, n), which determine the number of half-waves in the X-direction and of full waves in the Y-direction. Note that because of Equation 5.46, the prebuckling deformation of the shell is nonlinear. For numerical illustrations, two sets of material mixture are considered for an FGM cylindrical shell. One is silicon nitride and stainless steel (referred to as Si3N4=SUS304) and the other is zirconium oxide and titanium alloy (referred to as ZrO2=Ti-6Al-4V). The material properties are assumed to be nonlinear function of temperature of Equation 1.4, and typical values, in the present case, can be found in Tables 1.1 through 1.3. Poisson’s ratio is assumed to be a constant, that is, nf ¼ 0.28 for the Si3N4=SUS304 cylindrical shell, and nf ¼ 0.29 for the ZrO2=Ti-6Al-4V one. For these examples, the shell geometric parameter R=h ¼ 30, h ¼ 10 mm, and T0 ¼ 300 K (room temperature). It should be appreciated that in all ﬁgures W*=h ¼ 0:1 denotes the dimensionless maximum initial geometric imperfection of the shell. We ﬁrst examine the postbuckling load–shortening curves for an isotropic cylindrical shell with a local geometric imperfection subjected to axial compression. The results are compared in Figure 5.2 with the experimental

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163

100

Nx (N mm−1)

80

Shell A1a L = 340.0 mm, R = 199.9 mm, h = 0.509 mm (m, n) = (9, 17)

60

40

20

EXP: Hautala (1998) Present: W ∗/h = 0.25

0 0.0

0.2

0.4

0.6

0.8

1.0

∆x (mm)

FIGURE 5.2 Comparisons of postbuckling load– shortening curves for an isotropic cylindrical shell with a local geometric imperfection under axial compression.

results of Hautala (1998). The computing data adopted here are L ¼ 340 mm, R ¼ 199.9 mm, h ¼ 0.509 mm, E ¼ 190 GPa, and n ¼ 0.3. The results calculated show that when an initial geometric imperfection was present (W*=h ¼ 0:25), the limit point load Nx ¼ 71.63 N mm1, then the present results are in reasonable agreement with the experimental results. Tables 5.1 and 5.2 present buckling load P (in kN) for perfect, Si3N4=SUS304 and ZrO2=Ti-6Al-4V cylindrical shells with different values

TABLE 5.1 Comparisons of Buckling Loads Pcr (kN) for Perfect, Si3N4=SUS304 Cylindrical Shells Subjected to Axial Compression and under Thermal Environments (R=h ¼ 30, h ¼ 10 mm, T0 ¼ 300 K) Z

N¼0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

300

74942.67a

82424.49a

94842.38a

100394.40a

106562.10a

500

b

b

b

b

100443.90b

c

96739.29c

a

105298.20a

T-ID TU ¼ 300 K TL ¼ 300 K

800 TU ¼ 500 K

300

70484.06

c

68067.06

a

73175.55

77472.75

c

74876.27

a

80711.32

89258.90

c

86125.56

a

93379.08

94590.73 91151.51 99057.63

TL ¼ 300 K

500 800

67156.96 68185.81c

74226.40 74912.29c

86428.66 85843.08c

91970.67 90738.77c

97961.20b 96262.37c

TU ¼ 700 K

300

71132.21a

78753.39a

91745.87a

97575.29a

103913.40a

500

b

b

b

b

95442.02b

c

96743.75c

TL ¼ 300 K

800

b

b

b

b

63881.80

d

66541.07

71002.34

d

74116.27

83574.78

d

86708.57

89317.30 91453.34

(continued)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

TABLE 5.1 (continued) Comparisons of Buckling Loads Pcr (kN) for Perfect, Si3N4=SUS304 Cylindrical Shells Subjected to Axial Compression and under Thermal Environments (R=h ¼ 30, h ¼ 10 mm, T0 ¼ 300 K) Z

N¼0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

300 500

73175.55a 67156.96b

79927.86a 73504.35b

91263.13a 84416.09b

96301.30a 89316.01b

101806.70a 94572.83b

800

68185.81c

74188.52c

83967.88c

88318.46c

93208.09c

300

a

a

a

a

97348.63a

b

88957.43b

d

91594.48c

T-D (TU ¼ 500 K, TL ¼ 300 K) (TU ¼ 700 K, TL ¼ 300 K)

500 800

a b c d

71132.21

b

63881.80

d

66541.07

77422.60

b

69838.80

d

72870.63

87873.71

b

79904.80

d

82973.04

92449.94 84338.26 87234.89

Buckling mode (m, n) ¼ (4, 5). Buckling mode (m, n) ¼ (6, 5). Buckling mode (m, n) ¼ (8, 5). Buckling mode (m, n) ¼ (9, 6).

TABLE 5.2 Comparisons of Buckling Loads Pcr (kN) for Perfect, ZrO2=Ti-6Al-4V Cylindrical Shells Subjected to Axial Compression and under Thermal Environments (R=h ¼ 30, h ¼ 10 mm, T0 ¼ 300 K) Z

N¼0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

300

38137.02a

42208.14a

48959.04a

51989.80a

55358.61a

500

b

b

b

b

52068.91b

c

50681.81c

a

50900.87e

b

45465.31b

T-ID (TU ¼ 300 K, TL ¼ 300 K)

800 (TU ¼ 500 K, TL ¼ 300 K)

300 500

35788.11

c

34956.13

a

37625.46

b

34831.12

39583.18

c

38702.66

a

41193.16

b

37753.29

45977.06

c

44856.76

a

46802.43

b

42313.59

48877.66 47607.31 49126.52 44165.85

35032.66 37064.95a

c

39161.86 40030.25a

d

44039.69 44018.71e

d

(TU ¼ 700 K,

800 300

c

45793.84 44823.28e

47233.32d 44092.91e

TL ¼ 300 K)

500

33875.69b

35995.33b

39245.23b

40646.90b

42102.80b

800

d

35325.80

d

37744.23

d

41347.57

d

42934.16

45384.85d

300

37625.46a

39069.95a

40717.95a

40924.73a

39109.79a

500

b

b

b

b

36158.93b

d

39316.14d

a

41797.12a

e

14388.71b

b

25829.02b

T-D (TU ¼ 500 K, TL ¼ 300 K)

800 (TU ¼ 700 K, TL ¼ 300 K)

300 500 800

a b c d e

34831.12

c

35032.66

a

37064.95

b

33875.69

d

35325.80

Buckling mode (m, n) ¼ (4, 5). Buckling mode (m, n) ¼ (6, 5). Buckling mode (m, n) ¼ (8, 5). Buckling mode (m, n) ¼ (9, 6). Buckling mode (m, n) ¼ (5, 6).

35575.15

d

37173.96

a

35303.03

b

31333.34

d

33277.44

36249.34

d

38060.16

a

29385.48

b

34159.14

b

31713.80

36340.39 38349.19 26235.67 32257.93 27780.47

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Postbuckling of Shear Deformable FGM Shells

165

of volume fraction index N ( ¼ 0.0, 0.2, 1.0, 2.0, and 5.0) subjected to axial compression. Three thermal environmental conditions, referred to as I, II, and III, are considered. For case I, TU ¼ TL ¼ 300 K, for case II, TU ¼ 500 K, TL ¼ 300 K, and for case III, TU ¼ 700 K, TL ¼ 300 K. Here, T-D represents material properties for FGMs are temperature-dependent. T-ID represents material properties for FGMs are temperature-independent, i.e., in a ﬁxed temperature T0 ¼ 300 K, as previously used in Loy et al. (1999), Pradhan et al. (2000), and Ng et al. (2001). It can be seen that, for the Si3N4=SUS304 cylindrical shell, a fully metallic shell (N ¼ 0) has lowest buckling load and that the buckling load increases as the volume fraction index N increases. It can be seen that an increase is about þ39%, þ41%, and þ37%, respectively, for the Si3N4=SUS304 cylindrical shell with Z ¼ 300, 500, and 800 from N ¼ 0 to N ¼ 5 at the same thermal environmental condition II under T-D case. In contrast, for the ZrO2=Ti-6Al-4V cylindrical shell, the buckling load is lower than that of the Si3N4=SUS304 shell at the same loading conditions and erratic behavior can be observed under T-D case. It can also be seen that the temperature reduces the buckling load when the temperature dependency is put into consideration. The percentage decrease is about 7.9%, 10.8%, and 4.3% for the Si3N4=SUS304 cylindrical shell and about 49%, 34%, and 42% for the ZrO2=Ti-6Al-4V cylindrical shell with Z ¼ 300, 500, and 800 from thermal environmental condition case I to case III under the same volume fraction distribution N ¼ 2. In the following parametric study only T-D case is considered except for Figure 5.3. Typical results are shown in Figures 5.3 through 5.5 for the Si3N4=SUS304 cylindrical shell with z ¼ 500. Figure 5.3 gives the postbuckling load–shortening and load–deﬂection curves for a Si3N4=SUS304 cylindrical shell with volume fraction index N ¼ 2.0 subjected to axial compression under thermal environmental condition II and two cases of thermoelastic properties T-ID and T-D. It can be seen that the postbuckling equilibrium path becomes lower when the temperature-dependent properties are taken into account. Figure 5.4 presents the postbuckling load–shortening and load–deﬂection curves for the same cylindrical shell subjected to axial compression and under three thermal environmental conditions I, II, and III. It is found that an initial extension occurs as the temperature increases and the buckling loads are reduced with increases in temperature, and postbuckling path becomes lower. Figure 5.5 shows the effect of the volume fraction index N ( ¼ 0.2, 1.0, and 2.0) on the postbuckling behavior of Si3N4=SUS304 cylindrical shell subjected to axial compression and under thermal environmental condition II. The results show that the slope of the postbuckling load–shortening curve for the shell with N ¼ 2.0 are larger than others, and the shell has considerable postbuckling strength. From Figures 5.3 through 5.5, the well-known snap-through phenomenon could be found. The elastic limit loads for imperfect shells can be achieved and imperfection sensitivity can be predicted. The postbuckling

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166

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 150,000

100,000

Si3N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, TU = 500 K, TL = 300 K N = 2.0, (m, n) = (6, 5) I: T-ID I II: T-D

Px (kN)

II

50,000 W ∗/h = 0.0

0 −5

W ∗/h = 0.1

0

5

10 15 20 ∆x (mm)

(a)

25

30

35

150,000

Si3N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, TU = 500 K, TL = 300 K N = 2.0, (m, n) = (6, 5) I: T-ID 100,000 II: T-D Px (kN)

I II

I II

50,000

W ∗/h = 0.0 W ∗/h = 0.1

0 0.0 (b)

0.1

0.2

0.3

0.4

0.5

0.6

W/h

FIGURE 5.3 Effect of temperature dependency on the postbuckling behavior of a Si3N4=SUS304 cylindrical shell subjected to axial compression: (a) load shortening; (b) load deﬂection.

load–shortening and load–deﬂection curves for imperfect Si3N4=SUS304 cylindrical shells have been plotted, along with the perfect shell results, in Figures 5.3 through 5.5. Table 5.3 gives imperfection sensitivity l* for the imperfect Si3N4=SUS304 cylindrical shells with Z ¼ 300, 500, and 800,

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167

150,000

Px (kN)

Si3N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, N = 2.0, (m, n) = (6, 5) I: TU = TL = 300 K II: TU = 500 K, TL = 300 K 100,000 III: TU = 700 K, TL = 300 K I II III

50,000

W ∗/h = 0.0 W ∗/h = 0.1

0 −5

0

5

10

15

20

25

30

35

∆x (mm)

(a)

150,000

Si3N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, N = 2.0, (m, n) = (6, 5)

Px (kN)

100,000

I: TU = TL = 300 K II: TU = 500 K, TL = 300 K III: TU = 700 K, TL = 300 K III

II

I

50,000

W ∗/h = 0.0 W ∗/h = 0.1

0 0.0 (b)

0.1

0.2

0.3

0.4

0.5

0.6

W/h

FIGURE 5.4 Effect of temperature ﬁeld on the postbuckling behavior of a Si3N4=SUS304 cylindrical shell subjected to axial compression: (a) load shortening; (b) load deﬂection.

and with different values of volume fraction index N ( ¼ 0.2 and 2.0) subjected to axial compression and under thermal environmental conditions I and III. Here, l* is the maximum value of Px for the imperfect shell, made

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168

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 150,000

Si3N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, TU = 500 K, TL = 300 K (m, n) = (6, 5) 100,000 Px (kN)

I: N = 0.2 II: N = 1.0 III: N = 2.0

III II I

50,000

W ∗/h = 0.0 0

W ∗/h = 0.1 −5

0

(a)

5

10

15

20

25

30

35

∆x (mm)

150,000

Si3N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, TU = 500 K, TL = 300 K (m, n) = (6, 5)

Px (kN)

100,000

I

III II

I: N = 0.2 II: N = 1.0 III: N = 2.0

50,000

W ∗/h = 0.0 W ∗/h = 0.1

0 0.0

(b)

0.1

0.2

0.3

0.4

0.5

0.6

W/h

FIGURE 5.5 Effect of volume fraction index N on the postbuckling behavior of Si3N4=SUS304 cylindrical shells subjected to axial compression: (a) load shortening; (b) load deﬂection.

dimensionless by dividing by the critical value of Px for the perfect shell as shown in Table 5.1. These results show that the imperfection sensitivity becomes weak when temperature is increased and ﬁber volume fraction

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169

TABLE 5.3 Imperfection Sensitivity l* for Si3N4=SUS304 Cylindrical Shells Subjected to Axial Compression in Thermal Environments (R=h ¼ 30, h ¼ 10 mm, T0 ¼ 300 K) W*=h

Thermal Environmental Conditions

N

(TU ¼ 300 K,

0.2

TL ¼ 300 K)

0.0

0.05

0.1

0.15

0.20

300

1.0

0.776

0.653

0.564

0.495

500

1.0

0.774

0.641

0.548

0.479

800

1.0

0.777

0.641

0.548

—

300

1.0

0.773

0.649

0.560

0.490

500

1.0

0.770

0.636

0.543

0.475

800

1.0

0.775

0.638

0.545

—

0.2

300 500

1.0 1.0

0.812 0.842

0.683 0.698

0.590 0.597

— —

800

1.0

0.838

0.692

0.584

—

2.0

300

1.0

0.799

0.671

—

—

500

1.0

0.822

0.679

—

—

800

1.0

0.823

0.672

—

—

2.0

(TU ¼ 700 K, TL ¼ 300 K)

Z

only has small effect on the imperfection sensitivity. Note that for larger imperfection amplitudes, e.g., W*=h > 0.2, the postbuckling equilibrium path becomes stable, so that no imperfection sensitivity can be predicted.

5.4

Postbuckling Behavior of FGM Cylindrical Shells under External Pressure

Buckling under external pressure is another basic problem for cylindrical shells. We consider an FGM cylindrical shell subjected to external pressure q. Now the nonlinear differential equations in the Donnell sense are ~12 (Cx ) L ~13 (Cy ) þ L ~14 (F) L ~15 (N T ) L ~16 (MT ) 1 F,xx ~11 (W) L L R ~ ¼ L(W þ W*, F) þ q ~21 (F) þ L ~22 (Cx ) þ L ~23 (Cy ) L ~24 (W) L ~25 (N T ) þ 1 W,xx L R 1~ ¼ L(W þ 2W*, W) 2

(5:55)

(5:56)

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170

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells ~31 (W) þ L ~32 (Cx ) L ~33 (Cy ) þ L ~34 (F) L ~35 (N T ) L ~36 (ST ) ¼ 0 L

(5:57)

~41 (W) L ~42 (Cx ) þ L ~43 (Cy ) þ L ~44 (F) L ~45 (N T ) L ~46 (ST ) ¼ 0 L

(5:58)

Note that only Equation 5.55 has a small change with respect to Equation 5.15, whereas Equations 5.56 through 5.58 are identical in forms to those of Equations 5.16 through 5.18. The two end edges of the shell are assumed to be simply supported or clamped, in the present case, the boundary conditions are X ¼ 0, L: W ¼ Cy ¼ 0, Mx ¼ Px ¼ 0 W ¼ Cx ¼ Cy ¼ 0

(simply supported)

(5:59a)

(clamped)

(5:59b)

2pR ð

N x dY þ pR2 qa ¼ 0

(5:59c)

0

where a ¼ 0 and a ¼ 1 for lateral and hydrostatic pressure loading case, respectively. Also the closed (or periodicity) condition is expressed by Equation 5.20, and the average end-shortening relationship is deﬁned by Equation 5.21. Introducing dimensionless quantities of Equation 5.6, and * D22 * A11 * A22 * ]3=8 dq ¼ (Dx =L)(3)3=4 LR1=2 =4p[D11

(5:60)

The nonlinear Equations 5.55 through 5.58 may then be written in dimensionless form (boundary layer type) as «2 L11 (W) «L12 (Cx ) «L13 (Cy ) þ «g 14 L14 (F) g14 F,xx 4 ¼ g14 b2 L(W þ W*, F) þ g 14 (3)1=4 lq «3=2 3

(5:61)

L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) «g24 L24 (W) þ g 24 W,xx 1 ¼ g24 b2 L(W þ 2W*, W) 2 «L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) ¼ 0 «L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) ¼ 0

(5:62) (5:63) (5:64)

The boundary conditions expressed by Equation 5.59 become x ¼ 0, p: W ¼ Cy ¼ 0, Mx ¼ Px ¼ 0

(simply supported)

(5:65a)

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Postbuckling of Shear Deformable FGM Shells W ¼ Cx ¼ Cy ¼ 0 1 2p

2p ð

b2 0

171

(clamped)

(5:65b)

@2F 2 dy þ (3)1=4 lq «3=2 a ¼ 0 2 @y 3

(5:65c)

and the closed condition is expressed by Equation 5.29. It has been shown (Shen 2008b) that the effect of the boundary layer on the solution of a shell under external pressure is of the order «3=2, hence the unit end-shortening relationship may be written in dimensionless form as (3)3=4 3=2 « dq ¼ 2 8p g 24

2p ð ðp

g224 b2 0 0

@Cy @2F @2F @Cx þ g 233 b g5 2 þ g 24 g 511 @x @y @y2 @x

2 @2W 1 @W 2 2@ W «g 24 g611 þ g b g 244 @x2 @y2 2 24 @x 2

@W @W* g 24 þ « g 24 gT1 g 5 g T2 DT dx dy @x @x

(5:66)

Having developed the theory, we are now in a position to solve Equations 5.61 through 5.64 with boundary condition (Equation 5.65) by means of a singular perturbation technique. The essence of this procedure, in the present case, is to assume that ~ j, y, «) þ W(x, ^ z, y, «) W ¼ w(x, y, «) þ W(x, ~ j, y, «) þ F(x, ^ z, y, «) F ¼ f (x, y, «) þ F(x, ~ x (x, j, y, «) þ C ^ x (x, z, y, «) Cx ¼ cx (x, y, «) þ C

(5:67)

~ y (x, j, y, «) þ C ^ y (x, z, y, «) Cy ¼ cy (x, y, «) þ C in which the regular and boundary layer solutions are expressed in the forms of perturbation expansions as w(x, y, «) ¼

X

«j=2 wj=2 (x, y),

j¼1

cx (x, y, «) ¼

X

f (x, y, «) ¼

X j¼0

«j=2 (cx )j=2 (x, y),

cy (x, y, «) ¼

j¼1

~ j, y, «) ¼ W(x,

X X j¼0

X

«j=2 (cy )j=2 (x, y)

X

~j=2þ2 (x, j, y) «j=2þ2 F

j¼0 (jþ3)=2

«

(5:68a)

j¼1

~ j=2þ1 (x, j, y), F(x, ~ j, y, «) ¼ «j=2þ1 W

j¼0

~ x (x, j, y, «) ¼ C

«j=2 fj=2 (x, y)

~ x) ~ (C (jþ3)=2 (x, j, y), Cy (x, j, y, «) ¼

X

~ y) «j=2þ2 (C j=2þ2 (x, j, y)

j¼0

(5:68b)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

^ z, y, «) ¼ W(x,

X

^ j=2þ1 (x, z, y), F(x, ^ z, y, «) ¼ «j=2þ1 W

j¼0

^ x (x, z, y, «) ¼ C

X

X

^j=2þ2 (x, z, y) «j=2þ2 F

j¼0

«

(jþ3)=2

^ x) ^ (C (jþ3)=2 (x, z, y), Cy (x, z, y, «) ¼

j¼0

X

^ y) «j=2þ2 (C j=2þ2 (x, z, y)

j¼0

(5:68c)

We also let 4 1=4 3=2 X j (3) lq « ¼ « (ky )j 3 j¼0

(5:69)

Substituting Equation 5.67 into Equations 5.61 through 5.64 and collecting the terms of the same order of «, we can obtain three sets of perturbation equations for the regular and boundary layer solutions, respectively, e.g., the regular solutions w(x, y) and f(x, y) should satisfy O(«0 ): g14 ( f0 ),xx ¼ g14 b2 L(w0 , f0 ) þ ky0

(5:70a)

1 L21 ( f0 ) þ g 24 L22 (cx0 ) þ g24 L23 (cy0 ) þ g 24 (w0 ),xx ¼ g 24 b2 L(w0 , w0 ) (5:70b) 2 L32 (cx0 ) L33 (cy0 ) þ g14 L34 ( f0 ) ¼ 0 (5:70c) L42 (cx0 ) þ L43 (cy0 ) þ g14 L44 ( f0 ) ¼ 0

O(«1=2 ): g 14 ( f1=2 ),xx ¼ g 14 b2 L w1=2 , f0 þ L w0 , f1=2

(5:70d) (5:71a)

L21 ( f1=2 ) þ g24 L22 (cx(1=2) ) þ g 24 L23 (cy(1=2) ) þ g24 (w1=2 ),xx 1 ¼ g 24 b2 L(w0 , w1=2 ) 2 L32 (cx(1=2) ) L33 (cy(1=2) ) þ g 14 L34 ( f1=2 ) ¼ 0 L42 (cx(1=2) ) þ L43 (cy(1=2) ) þ g 14 L44 ( f1=2 ) ¼ 0

(5:71b) (5:71c) (5:71d)

O(«1 ): L12 (cx0 ) L13 (cy0 ) þ g 14 L14 ( f0 ) g14 ( f1 ),xx ¼ g14 b2 [Lðw1 , f0 Þ þ Lðw0 , f1 Þ] þ ky1

(5:72a)

L21 ( f1 ) þ g24 L22 (cx1 ) þ g 24 L23 (cy1 ) g24 L24 (w0 ) þ g24 (w1 ),xx 1 ¼ g 24 b2 L(w0 , w1 ) 2 L31 (w0 ) þ L32 (cx1 ) L33 (cy1 ) þ g14 L34 ( f1 ) ¼ 0 L41 (w0 ) L42 (cx1 ) þ L43 (cy1 ) þ g14 L44 ( f1 ) ¼ 0

(5:72b) (5:72c) (5:72d)

O(«3=2 ): L12 (cx(1=2) ) L13 (cy(1=2) ) þ g14 L12 ( f1=2 ) g 14 ( f3=2 ),xx ¼ g 14 b2 [L(w3=2 , f0 ) þ L(w1 , f1=2 ) þ L(w1=2 , f1 ) þ L(w0 , f3=2 )]

(5:73a)

L21 ( f3=2 ) þ g 24 L22 (cx(3=2) ) þ g24 L23 (cy(3=2) ) g24 L22 (w1=2 ) þ g 24 (w3=2 ),xx 1 ¼ g 24 b2 [L(w1=2 , w1 ) þ L(w3=2 , w0 )] 2

(5:73b)

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173

L31 (w1=2 ) þ L32 (cx(3=2) ) L33 (cy(3=2) ) þ g14 L34 ( f3=2 ) ¼ 0

(5:73c)

L41 (w1=2 ) L42 (cx(3=2) ) þ L43 (cy(3=2) ) þ g 14 L44 ( f3=2 ) ¼ 0

(5:73d)

O(«2 ): L11 (w0 ) L12 (cx1 ) L13 (cy1 ) þ g14 L12 ( f1 ) g 14 ( f2 ),xx ¼ g14 b2 [L(w2 þ W*, f0 ) þ L(w3=2 , f1=2 ) þ L(w1 , f1 ) þ L(w1=2 , f3=2 ) þ L(w0 , f2 )] þ ky2

(5:74a)

L21 ( f2 ) þ g 24 L22 (cx2 ) þ g24 L23 (cy2 ) g 24 L24 (w1 ) þ g 24 (w2 ),xx 1 ¼ g24 b2 [L(w1 , w1 ) þ L(w3=2 , w1=2 ) þ Lðw2 þ 2W*, w0 Þ] 2

(5:74b)

L31 (w1 ) þ L32 (cx2 ) L33 (cy2 ) þ g 14 L34 ( f2 ) ¼ 0

(5:74c)

L41 (w1 ) L42 (cx2 ) þ L43 (cy2 ) þ g 14 L44 ( f2 ) ¼ 0

(5:74d) (3=2)

From Equation 5.68a, we assume that w0 ¼ w1=2 ¼ w1 ¼ 0 and w3=2 ¼ A00 , along with cx0 ¼ cx(1=2) ¼ cx1 ¼ cx(3=2) ¼ 0, cy0 ¼ cy(1=2) ¼ cy1 ¼ cy(3=2) ¼ 0, (1) 2 2 2 2 2 2 f1=2 ¼ f3=2 ¼ 0, f0 ¼ B(0) 00 (b x þ ay =2), and f1 ¼ B00 (b x þ ay =2), from 2 (1) Equations 5.70a and 5.72a we have ky0 ¼ b2 B(0) 00 and ky1 ¼ b B00 . The initial buckling mode is assumed to have the form

w2 (x, y) ¼ A(2) 11 sin mx sin ny

(5:75)

and the initial geometric imperfection is assumed to have the similar form * sin mx sin ny ¼ «2 mA(2) W*(x, y, «) ¼ «2 a11 11 sin mx sin ny

(5:76)

* =A(2) where m ¼ a11 11 is the imperfection parameter. From Equation 5.74 one has 1 2 2 2 þ B(2) ay b x þ f2 ¼ B(2) 00 11 sin mx sin ny 2 cx2 ¼ C(2) 11 cos mx sin ny cy2 ¼

D(2) 11

(5:77)

sin mx cos ny

and ky2 ¼ b2 B(2) 00 , D(2) 11 ¼ g 14 nb

B(2) 11 ¼

g02 (2) B , g00 11

g24 m2 (2) A11 , g06 b2 B(0) 00 ¼

C(2) 11 ¼ g 14 m

g01 (2) B , g00 11

g24 m4 (1 þ m)(n2 b2 þ (1=2) am2 )g06

(5:78)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

~ ~ For boundary layer solutions, W(x, j, y) and F(x, j, y) should satisfy boundary layer equations of each order, for example O(«5=2 ): g110 ~5=2 @4F @j4

~ 3=2 @4W @j

g120

4

~5=2 ~5=2 ~ x2 @4F @2F @3C þ g14 g 140 g14 ¼0 3 4 @j @j @j2

~ 3=2 ~ 3=2 ~ x2 @4W @2W @3C g g þ g ¼0 24 240 24 @j3 @j4 @j2 ~ 3=2 ~5=2 ~ x2 @3W @3F @2C g þ g g ¼0 g310 320 14 220 @j3 @j2 @j3 ~ y(5=2) @2C ¼0 g 430 @j2

þ g24 g 220

(5:79a) (5:79b) (5:79c) (5:79d)

which leads to ~ 3=2 @4W @j

4

þ 2c

~ 3=2 @2W @j2

~ 3=2 ¼ 0 þ b2 W

(5:80)

where c and b are deﬁned by Equation 5.43. The solution of Equation 5.80 can be written as ~ 3=2 ¼ a(3=2) sin fj þ a(3=2) cos fj eqj (5:81a) W 10 01 and

~5=2 ¼ b(5=2) sin fj þ b(5=2) cos fj eqj F 10 01 ~ x2 ¼ c(2) sin fj þ c(2) cos fj eqj C 10 01 ~ y(5=2) ¼ 0 C

(5:81b) (5:81c) (5:81d)

where q and f are deﬁned by Equation 5.45. ^ ^ Similarly, the boundary layer solutions W(x, z, y) and F(x, z, y) can be obtained in the same manner. Then matching the regular solutions with the boundary layer solutions at ~ 3=2 ) ¼ (w3=2 þ W ~ 3=2 ),x ¼ 0 and the each end of the shell, e.g., let (w3=2 þ W ^ ^ (w3=2 þ W3=2 ) ¼ (w3=2 þ W3=2 ),x ¼ 0 at x ¼ 0 and x ¼ p edges, respectively, so that the asymptotic solutions satisfying the clamped boundary conditions are constructed as

x x x (3=2) (3=2) (3=2) (3=2) W ¼ «3=2 A00 A00 a01 cos f pﬃﬃﬃ þ a10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « « px px px (3=2) (3=2) (3=2) a01 cos f pﬃﬃﬃ þ a10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ A00 « « « h i h i (2) (3) 2 3 þ « A11 sin mx sin ny þ « A11 sin mx sin ny h i (4) (4) (4) þ A sin mx sin ny þ A cos 2mx þ A cos 2ny þ O(«5 ) þ «4 A(4) 00 11 20 02

(5:82)

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Postbuckling of Shear Deformable FGM Shells

175

1 (0) 2 2 y2 1 (1) 2 2 y2 þ « B00 b x þ a F ¼ B00 b x þ a 2 2 2 2

2 1 (2) 2 2 y (2) 2 þ B11 sin mx sin ny þ « B00 b x þ a 2 2

x x x (3=2) (5=2) (5=2) b01 cos f pﬃﬃﬃ þ b10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ þ «5=2 A00 « « « p x p x p x (3=2) (5=2) (5=2) þ A00 b01 cos f pﬃﬃﬃ þ b10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « «

2 2 1 y 1 y þ «3 B(3) þ «4 B(4) b2 x 2 þ a b2 x 2 þ a 2 2 2 00 2 00 i (4) (4) 5 þ B20 cos 2mx þ B02 cos 2ny þ O(« )

(5:83)

x x x (2) (2) Cx ¼ «2 C(2) 11 cos mx sin ny þ c01 cos f pﬃﬃﬃ þ c10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « « p x p x p x (2) exp q pﬃﬃﬃ þ c(2) 01 cos f pﬃﬃﬃ þ c10 sin f pﬃﬃﬃ « « « h i h i (3) (4) 3 4 5 þ « C11 cos mx sin ny þ « C11 cos mx sin ny þ C(4) 20 sin 2mx þ O(« )

(5:84)

h i h i (3) 3 Cy ¼ «2 D(2) 11 sin mx cos ny þ « D11 sin mx cos ny h i (4) þ «4 D(4) sin mx cos ny þ D sin 2ny þ O(«5 ) 11 02

(5:85)

It can be seen that the solutions expressed by Equations 5.82 through 5.85 are quite different from those of the shell subjected to axial compression. As in the case of axial compression, all of the coefﬁcients in Equations 5.82 through 5.85 are related and can be expressed in terms of A(2) 11 , as shown in (j) , except for A (j ¼ 3=2, 2, 3, 4) which may be deterEquation 5.78 for B(2) 00 11 mined by substituting Equations 5.82 through 5.85 into closed condition, and can be written as 1 1 4 1=4 1 (3=2) (3) lq «1=2 1 ag 5 [g g 5 g T1 ]DT, A00 ¼ g24 2 3 g24 T2 (5:86) 2 1 2 2 (3) (4) (2) ¼ A ¼ 0, A ¼ b (1 þ 2m) A A(2) n 00 00 00 11 8 Next, upon substitution of Equations 5.82 through 5.85 into the boundary condition (Equation 5.65c) and into Equation 5.66, the postbuckling equilibrium paths can be written as

2 1 (2) 2 (2) þ l A « þ (5:87) lq ¼ (3)3=4 «3=2 l(0) q q 11 4 and 2 (T) (2) 2 dq ¼ d(0) A(2) þ x dx þ dx 11 «

(5:88)

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

2 In Equations 5.87 and 5.88, A(2) « is taken as the second perturbation 11 parameter relating to the dimensionless maximum deﬂection. From Equation 5.82, by taking (x, y) ¼ (p=2m, p=2n), one has 2 2 A(2) 11 « ¼ Wm Q4 Wm þ

(5:89)

where Wm is the dimensionless form of maximum deﬂection of the shell that can be written as " # 1 h W þ Q3 « Wm ¼ C3 ½D11 * D22 * A11 * A22 * 1=4 h

(5:90)

All symbols used in Equations 5.87 through 5.90 are described in detail in Appendix M. The perturbation scheme described here is quite different from that of traditional one (Budiansky and Amazigo 1968, Amazigo 1974). As argued before, the traditional asymptotic solutions are only suitable for initial postbuckling analysis and cannot be extended to the deep postbuckling range. Equations 5.87 through 5.90 can be employed to obtain numerical results for full nonlinear postbuckling load–shortening and load–deﬂection curves of FGM cylindrical shells subjected to external pressure. For numerical illustrations, two sets of material mixture, as shown in Section 5.3, for FGM cylindrical shells are considered. We ﬁrst examine the buckling loads for simply supported, isotropic cylindrical shells under hydrostatic pressure. The results are calculated and compared in Table 5.4 with classical shell theory results of Hutchinson and Amazigo (1967), and ﬁnite element results obtained by Kasagi and Sridharan (1993). The material properties used in this case, as given in Hutchinson and Amazigo (1967), are E ¼ 10 106 psi, and n ¼ 0.33. It can be seen that, for most cases the present results agree well with existing results. In contrast, for very short cylinders (ZB ¼ 10), the present results are lower than those of Hutchinson and Amazigo (1967) and Kasagi and Sridharan (1993). This is because in the present analysis the nonlinear prebuckling deformations are considered, and the effect of the boundary layer is great for very short cylinders. In addition, the postbuckling load–deﬂection curves for an isotropic interring short cylindrical shell (model no. 7) subjected to lateral pressure are compared in Figure 5.6 with the experimental results of Seleim and Roorda (1987). The computing data adopted here are L ¼ 3.5 in., R ¼ 5.04 in., h ¼ 0.08 in., and n ¼ 0.3. Then the postbuckling load–deﬂection curves for an isotropic long cylindrical shell subjected to lateral pressure are compared in Figure 5.7 with the experimental and FEM results of Djerroud et al. (1991). The computing data adopted here are L ¼ 150 mm, R ¼ 75 mm, h ¼ 0.16 mm, E ¼ 166.473 GPa, and n ¼ 0.34. The results calculated show that when an initial geometric imperfection was present, e.g., W*=h ¼ 0.01 for Figure 5.6 and W*=h ¼ 0.1 for Figure 5.7, the present results are in reasonable agreement with the experimental results.

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177

TABLE 5.4 Comparisons of Buckling Loads qcr (psi) for Perfect Isotropic Cylindrical Shells under Hydrostatic Pressure (E ¼ 10 106 psi, n ¼ 0.33) R=h

ZB

50

200

a

Hutchinson and Amazigo (1967)

Present HSDT

Kasagi and Sridharan FEM (1993)

10

1383.623 (1, 9)a

1425.0 (1, 9)

1390.0 (1, 9)

50

566.085 (1, 7)

570.2 (1, 7)

560.0 (1, 7)

100

389.620 (1, 6)

391.8 (1, 6)

385.6 (1, 6)

500

166.770 (1, 4)

167.3 (1, 4)

165.0 (1, 4)

1,000 5,000

124.984 (1, 3) 56.566 (1, 2)

125.1 (1, 3) 56.6 (1, 2)

123.5 (1, 3) 55.9 (1, 2)

10,000

37.022 (1, 2)

37.1 (1, 2)

36.65 (1, 2)

10

87.077 (1, 17)

89.07 (1, 18)

88.65 (1, 18)

50

35.167 (1, 13)

35.25 (1, 13)

35.09 (1, 13)

100

24.305 (1, 11)

24.35 (1, 11)

24.26 (1, 11)

500

10.436 (1, 8)

10.45 (1, 8)

10.42 (1, 8)

1,000

7.398 (1, 7)

7.412 (1, 7)

7.388 (1, 7)

5,000 10,000

3.416 (1, 5) 2.315 (1, 4)

3.423 (1, 5) 2.319 (1, 4)

3.412 (1, 5) 2.312 (1, 4)

The numbers in brackets indicate the buckling mode (m, n).

Tables 5.5 and 5.6 present buckling pressure qcr (in kPa) for perfect, Si3N4=SUS304 and ZrO2=Ti-6Al-4V cylindrical shells with different values of volume fraction index N ( ¼ 0.0, 0.2, 1.0, 2.0, and 5.0) subjected to lateral pressure. As in the case of axial compression, three thermal 2.0

q/qcr

1.5

Lateral pressure isotropic cylindrical shell R/h = 63, Z = 30.38 (m, n) = (1, 9)

1.0

W ∗/h = 0.0

0.5

W ∗/h = 0.01 EXP: Seleim and Roorda (1987)

0.0 0.0

0.5

1.0 W/h

1.5

FIGURE 5.6 Comparisons of postbuckling load–deﬂection curves for an isotropic short cylindrical shell under lateral pressure.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 1.6

Lateral pressure isotropic cylindrical shell (n = 0.34) R/h = 468.75, Z = 1875 (m, n) = (1, 9)

1.4 1.2 q/qcr

1.0 0.8

Present

0.6

W ∗/h = 0.1 Djerroud et al. (1991) EXP FEM

0.4 0.2 0.0

2

0

4

6

8

10

W/h FIGURE 5.7 Comparisons of postbuckling load–deﬂection curves for an isotropic long cylindrical shell under lateral pressure.

TABLE 5.5 Comparisons of Buckling Loads qcr (kPa) for Perfect, Si3N4=SUS304 Cylindrical Shells Subjected to Lateral Pressure and under Thermal Environments (R=h ¼ 30, h ¼ 10 mm, T0 ¼ 300 K) Z

N¼0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

300

13945.96a

15425.75a

17541.49a

18383.41a

19426.02a

500

b

b

b

b

14103.74b

b

11126.91b

a

19425.95a

b

14103.71b

T-ID (TU ¼ 300 K, TL ¼ 300 K)

800 (TU ¼ 500 K, TL ¼ 300 K)

300 500

10026.28

b

7969.70

a

13945.88

b

10026.25

11060.52

b

8809.92

a

15425.66

b

11060.49

12648.07

b

10031.34

a

17541.41

b

12648.04

13318.71 10524.54 18383.33 13318.68

7969.69 13945.90a

b

8809.91 15425.66a

b

10031.33 17541.38a

b

(TU ¼ 700 K,

800 300

b

10524.53 18383.30a

11126.90b 19425.91a

TL ¼ 300 K)

500

10026.28b

11060.51b

12648.04b

13318.67b

14103.70b

800

7969.72b

8809.93b

10031.34b

10524.53b

11126.90b

300

13945.88a

15270.61a

17170.87a

17919.19a

18839.65a

500

b

b

b

b

13666.14b

b

10788.86b

a

18360.08a

b

13308.43b

b

10512.42b

T-D (TU ¼ 500 K, TL ¼ 300 K)

800 (TU ¼ 700 K, TL ¼ 300 K)

300 500 800

a b

10026.25

b

7969.69

a

13945.90

b

10026.28

b

7969.72

Buckling mode (m, n) ¼ (1, 4). Buckling mode (m, n) ¼ (1, 3).

10951.44

b

8721.74

a

15143.17

b

10861.94

b

8649.31

12374.80

b

9818.30

a

16865.60

b

12150.18

b

9642.90

12971.41 10256.76 17538.02 12686.65 10036.95

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Postbuckling of Shear Deformable FGM Shells

179

TABLE 5.6 Comparisons of Buckling Loads qcr (kPa) for Perfect, ZrO2=Ti-6Al-4V Cylindrical Shells Subjected to Lateral Pressure and under Thermal Environments (R=h ¼ 30, h ¼ 10 mm, T0 ¼ 300 K) Z

N¼0

N ¼ 0.2

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

300 500

7133.13a 5122.51b

7941.64a 5687.28b

9095.10a 6552.62b

9557.11a 6919.87b

10131.67a 7351.27b

800

4075.35b

4534.36b

5200.24b

5470.73b

5802.46b

300

a

a

a

a

10131.70a

b

7351.34b

b

5802.50b

a

10132.36a

b

7351.80b

b

T-ID (TU ¼ 300 K, TL ¼ 300 K) (TU ¼ 500 K, TL ¼ 300 K)

500 800

(TU ¼ 700 K, TL ¼ 300 K)

300 500

7133.10

b

5122.50

b

4075.34

a

7133.09

b

5122.49

b

7941.59

b

5687.27

b

4534.36

a

7941.60

b

5687.29

b

9095.05

b

6552.62

b

5200.24

a

9095.20

b

6552.74

b

9557.07 6919.88 5470.74 9557.36 6920.09

800

4075.34

4534.37

5200.30

5470.85

5802.72b

300

7133.10a

7582.31a

8233.86a

8482.65a

8781.91a

500

b

b

b

b

6345.85b

b

5024.47b

a

8057.13a

b

5805.49b

b

4602.22b

T-D (TU ¼ 500 K, TL ¼ 300 K)

800 (TU ¼ 700 K, TL ¼ 300 K)

300 500 800

a b

5122.50

b

4075.34

a

7133.09

b

5122.49

b

4075.34

5435.24

b

4330.18

a

7381.57

b

5295.18

b

4216.22

5920.03

b

4705.55

a

7747.77

b

5566.09

b

4426.50

6118.53 4851.32 7885.16 5675.56 4506.44

Buckling mode (m, n) ¼ (1, 4). Buckling mode (m, n) ¼ (1, 3).

environmental conditions, referred to as I, II, and III, are considered. It can be found that the buckling pressure of ZrO2=Ti-6Al-4V cylindrical shell is lower than that of Si3N4=SUS304 cylindrical shell at the same loading conditions. It can be seen that a fully metallic shell (N ¼ 0) has lowest buckling load and that the buckling pressure increases as the volume fraction index N increases. It can be seen that an increase is about þ36% for the Si3N4=SUS304 cylindrical shell and about þ24% for the ZrO2=Ti-6Al-4V cylindrical shell from N ¼ 0 to N ¼ 5 at the same thermal environmental condition II under T-D case. It can also be seen that the temperature reduces the buckling pressure when the temperature dependency is put into consideration. The percentage decrease is about 4.7% for the Si3N4=SUS304 cylindrical shell and about 18% for the ZrO2=Ti-6Al-4V cylindrical shell from thermal environmental condition case I to case III under the same volume fraction distribution N ¼ 2. Figures 5.8 through 5.10 are postbuckling results for the same Si3N4=SUS304 cylindrical shell analogous to the compressive postbuckling results of Figures 5.3 through 5.5, which are for the loading case of hydrostatic pressure under environmental conditions. It can be seen that an increase in pressure is usually required to obtain an increase in deformation, and the postbuckling path is stable for both perfect and imperfect shells, and the shell structure is virtually imperfection-insensitive. This conclusion

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C005 Final Proof page 180 9.12.2008 4:42pm Compositor Name: VBalamugundan

180

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 20,000

Si3/N4/SUS304 R/h = 30, Z = 500, N = 2.0 T0 = 300 K, TU = 500 K, TL = 300 K (m, n) = (1, 3) I

15,000

q (kPa)

II 10,000 I: T-ID II: T-D 5,000 W ∗/h = 0.0 W ∗/h = 0.1

0 −4

−2

0

2

4

6

8

∆x (mm)

(a)

20,000

Si3/N4/SUS304 R/h = 30, Z = 500, N = 2.0

T0 = 300 K, TU = 500 K, TL = 300 K

15,000 I

(m, n) = (1, 3)

q (kPa)

II 10,000 I: T-ID II: T-D

5,000

W ∗/h = 0.0 W ∗/h = 0.1

0 (b)

0

2

4

6

8

W/h

FIGURE 5.8 Effect of temperature dependency on the postbuckling behavior of a Si3N4=SUS304 cylindrical shell subjected to hydrostatic pressure: (a) load shortening; (b) load deﬂection.

was validated recently in molecular dynamics simulation tests for carbon nanotubes subjected to external pressure (Zhang and Shen 2006). To compare Figures 5.5 and 5.10, it can be seen that now the slope of the postbuckling load–shortening curve for the shell with these three values of

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Postbuckling of Shear Deformable FGM Shells

181

20,000 Si3/N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, N = 2.0 (m, n) = (1, 3)

q (kPa)

15,000

10,000

I

II

III

I: TU = TL = 300 K II: TU = 500 K, TL = 300 K III: TU = 700 K, TL = 300 K

5,000 W ∗/h = 0.0 W ∗/h = 0.1

0

−4

−2

0

2

4

6

8

∆x (mm)

(a)

20,000 Si3/N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, N = 2.0 (m, n) = (1, 3)

q (kPa)

15,000

I II III 10,000

I: TU = TL = 300 K II: TU = 500 K, TL = 300 K III: TU = 700 K, TL = 300 K

5,000

W ∗/h = 0.0 W ∗/h = 0.1

0 (b)

0

2

4

6

8

W/h

FIGURE 5.9 Effect of temperature ﬁeld on the postbuckling behavior of a Si3N4=SUS304 cylindrical shell subjected to hydrostatic pressure: (a) load shortening; (b) load deﬂection.

volume fraction index N ( ¼ 0.2, 1.0, and 2.0) is almost the same, and the shell with N ¼ 2 has higher buckling pressure and considerable postbuckling strength. Otherwise, they lead to broadly the same conclusions as do Figures 5.3 and 5.4.

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C005 Final Proof page 182 9.12.2008 4:42pm Compositor Name: VBalamugundan

182

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 20,000 Si3/N4/SUS304 R/h = 30, Z = 500

T0 = 300 K, TU = 500 K, TL = 300 K

15,000

(m, n) = (1, 3)

q (kPa)

III II I

10,000

I: N = 0.2 II: N = 1.0 III: N = 2.0

5,000

W ∗/h = 0.0 W ∗/h = 0.1

0 −4

−2

0

2

4

6

8

∆x (mm)

(a)

20,000 Si3/N4/SUS304 R/h = 30, Z = 500 T0 = 300 K, TU = 500 K, TL = 300 K

15,000

(m, n) = (1, 3) III

q (kPa)

II I

10,000

I: N = 0.2 II: N = 1.0 III: N = 2.0

5,000

W ∗/h = 0.0 W ∗/h = 0.1

0

(b)

0

2

4

6

8

W/h

FIGURE 5.10 Effect of volume fraction index N on the postbuckling behavior of Si3N4=SUS304 cylindrical shells subjected to hydrostatic pressure: (a) load shortening; (b) load deﬂection.

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C005 Final Proof page 183 9.12.2008 4:42pm Compositor Name: VBalamugundan

Postbuckling of Shear Deformable FGM Shells

5.5

183

Postbuckling Behavior of FGM Cylindrical Shells under Torsion

Torsional buckling is an unsolved basic problem for cylindrical shells. This is due to the fact that the solution becomes more complicated in the case of cylindrical shells under torsion. The key issue is how to model the buckling mode of the shell under torsion. Loo (1954) and Nash (1957) suggested solutions formed as W ¼ W1 sin (pX=L) sin (nY=R þ knX=R)

(5:91)

W ¼ W1 [1 cos (2pX=L)] sin (nY=R þ knX=R)

(5:92)

where the parameter k is determined by minimizing the strain energy. Equation 5.91 was also adopted by Soﬁyev and Schnak (2004) for dynamic stability analysis of FGM cylindrical shells under torsional loading. It is worthy to note that both Equations 5.91 and 5.92 cannot satisfy boundary conditions such as simply supported or clamped at the end of the cylindrical shell and can be as approximate solutions. Yamaki and Matsuda (1975) attempted to give more accurate solutions as W¼

XX

Wmn (Cm1,n þ Cmþ1,n )

(5:93a)

m¼1 n¼0

Cmn ¼ cos (mx ny) þ (1)m cos (mx ny)

(5:93b)

Since sufﬁcient numbers of unknown parameters are retained, the solutions of Equation 5.93 could satisfy both compatibility and boundary conditions, but they do not satisfy equilibrium equation, and therefore, the Galerkin method had to be performed. Therefore, we need a clear understanding of the mechanisms of cylindrical shells subjected to torsion. We assume an FGM cylindrical shell subjected to a torque uniformly applied along the edges. Now the nonlinear differential equations in the Donnell sense are the same as in the case of axial compression, i.e., Equations ~ij() and nonlinear operator Lij() 5.15 through 5.18, where all linear operators L are deﬁned by Equation 1.37, and the forces, moments, and higher order moments caused by elevated temperature are deﬁned by Equation 1.28. The two end edges of the shell are assumed to be simply supported or clamped, in the present case the boundary conditions are X ¼ 0, L: W ¼ Cy ¼ 0,

Mx ¼ Px ¼ 0 (simply supported)

W ¼ Cx ¼ Cy ¼ 0

(clamped)

(5:94a) (5:94b)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C005 Final Proof page 184 9.12.2008 4:42pm Compositor Name: VBalamugundan

184

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 2pR ð

N xy dY MS ¼ 0

R

(5:94c)

0

where MS ¼ 2pR2 ht s and t s is the shear stress. Also the closed (or periodicity) condition is expressed by Equation 5.20, and the average end-shortening relationship is deﬁned by Equation 5.21. The angle of twist is deﬁned as 1 ¼ L

! ðL ðL " @U @V 1 @2F 4 @Cx @Cy * * 2 E66 * þ A66 þ dX ¼ B66 @Y @X @Y @X L @X@Y 3h 0

þ

0

4 @2W @W @W @W @W* @W @W* * 2E þ þ þ dX 66 3h2 @X@Y @X @Y @X @Y @Y @X

(5:95)

Introducing dimensionless quantities of Equation 5.6, and * (4=3h2 )E66 * =[D11 * D22 * A11 * A22 * ]1=4 g 566 ¼ B66 * =[D11 * D22 * A11 * A22 * ]1=4 g 666 ¼ (4=3h2 )E66 * D22 * A11 * A22 * ]3=16 =p1=2 [D11 * D22 * ]1=2 ls ¼ t s L1=2 R3=4 h[D11

(5:96)

* D22 * A11 * A22 * ]5=16 (ds , g) ¼ (Dx =L, G)L1=2 R3=4 =p1=2 [D11 The boundary layer-type equations may then be written in the same forms as in the case of axial compression, i.e., «2 L11 (W) «L12 (Cx ) «L13 (Cy ) þ «g 14 L14 (F) g14 F,xx ¼ g14 b2 L(W þ W*, F) (5:97)

L21 (F) þ g 24 L22 (Cx ) þ g24 L23 (Cy ) «g24 L24 (W) þ g 24 W,xx 1 ¼ g24 b2 L(W þ 2W*, W) 2

(5:98)

«L31 (W) þ L32 (Cx ) L33 (Cy ) þ g 14 L34 (F) ¼ 0

(5:99)

«L41 (W) L42 (Cx ) þ L43 (Cy ) þ g 14 L44 (F) ¼ 0

(5:100)

The boundary conditions expressed by Equation 5.91 become x ¼ 0, p: W ¼ Cy ¼ 0, Mx ¼ Px ¼ 0

(simply supported)

W ¼ Cx ¼ Cy ¼ 0

(clamped)

(5:101a) (5:101b)

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Postbuckling of Shear Deformable FGM Shells

1 2p

2p ð

b 0

@2F dy þ ls «5=4 ¼ 0 @x@y

185

(5:101c)

and the closed condition is expressed by Equation 5.29. It has been shown (Shen 2008c) that the effect of the boundary layer on the solution of a shell under torsion is of the order «5=4, the unit end-shortening relationship may be written in dimensionless form as 1 «5=4 ds ¼ 2 2p g 24

2p ð ðp

g224 b2 0 0

@Cy @2F @2F @Cx þ g þ g g g b 5 24 511 233 @x @y @y2 @x2

2 @ W 2@ W þ g b «g24 g611 244 @x2 @y2

þ « g224 g T1 g 5 gT2 DT dx dy 2

1 @W 2 @W @W* g 24 g 24 2 @x @x @x (5:102)

and the angle of twist may be written in dimensionless form as 1 5=4 g¼ « pg24

ðp g266 b 0

@2F @Cx @Cy þ g24 g566 b @y @x @x@y

@2W @W @W @W @W* @W @W* þ þ þ «g24 2g666 b þ g24 b dx @x@y @x @y @y @x @x @y

(5:103)

T

Note that N xy is zero-valued for an FGM cylindrical shell, so that no thermal effect remains in Equation 5.103. Having developed the theory, we are now in a position to solve Equations 5.97 through 5.100 with boundary condition (Equation 5.101) by means of a singular perturbation technique. The essence of this procedure, in the present case, is to assume that ~ j, y, «) þ W(x, ^ z, y, «) W ¼ w(x, y, «) þ W(x, ~ j, y, «) þ F(x, ^ z, y, «) F ¼ f (x, y, «) þ F(x, ~ x (x, j, y, «) þ C ^ x (x, z, y, «) Cx ¼ cx (x, y, «) þ C

(5:104)

~ y (x, j, y, «) þ C ^ y (x, z, y, «) Cy ¼ cy (x, y, «) þ C in which the regular and boundary layer solutions are expressed in the forms of perturbation expansions as

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186

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

w(x, y, «) ¼

X

«j=4þ1 wj=4þ1 (x, y),

f (x, y, «) ¼

j¼1

cx (x, y, «) ¼

X

X j¼0

«j=4þ1 (cx )j=4þ1 (x, y), cy (x, y, «) ¼

«j=4 fj=4 (x, y) X

j¼1

«j=4þ1 (cy )j=4þ1 (x, y)

j¼1

(5:105a) ~ j,y,«) ¼ W(x,

X

~ j=4þ1 (x, j,y), F(x,j,y, ~ «j=4þ1 W «) ¼

X

j¼1

~ x (x, j,y,«) ¼ C

X

~j=4þ2 (x,j, y) «j=4þ2 F

j¼1 j=4þ3=2

«

~ x) ~ (C j=4þ3=2 (x, j,y), Cy (x,j,y, «) ¼

X

j¼1

~ y) «j=4þ2 (C j=4þ2 (x, j,y)

j¼1

(5:105b)

^ z, y,«) ¼ W(x,

X

^ j=4þ1 (x,z,y), F(x, ^ z, y,«) ¼ «j=4þ1 W

j¼1

^ x (x, z, y, «) ¼ C

X

X

^j=4þ2 (x, z, y) «j=4þ2 F

j¼1

«

j=4þ3=2

^ x) ^ (C j=4þ3=2 (x,z,y), Cy (x, z, y,«) ¼

j¼1

X

^ y) «j=4þ2 (C j=4þ2 (x, z, y)

j¼1

(5:105c)

Substituting Equation 5.104 into Equations 5.97 and 5.100 and collecting the terms of the same order of «, we can obtain three sets of perturbation equations for the regular and boundary layer solutions, respectively. From Equation 5.105a, we assume that w0 ¼ w1=4 ¼ w1=2 ¼ w3=4 ¼ w1 ¼ 0, cx0 ¼ cx(1=2) ¼ cx(1=2) ¼ cx(3=4) ¼ cx1 ¼ 0, cy0 ¼ cy(1=4) ¼ cy(1=2) ¼ cy(3=4) ¼ cy1 ¼ 0, then the regular solutions w(x, y) and f(x, y) should satisfy O(«0 ): g14 ( f0 ),xx ¼ 0

(5:106a)

L21 ( f0 ) ¼ 0

(5:106b)

g 14 L34 ( f0 ) ¼ 0

(5:106c)

g 14 L44 ( f0 ) ¼ 0

(5:106d)

O(«5=4 ): g 14 ( f5=4 ),xx ¼ g 14 b2 [L(w5=4 , f0 )]

(5:107a)

L21 ( f5=4 ) þ g 24 L22 (cx(5=4) ) þ g24 L23 (cy(5=4) ) þ g24 (w5=4 ),xx ¼ 0

(5:107b)

L32 (cx(5=4) ) L33 (cy(5=4) ) þ g 14 L34 ( f5=4 ) ¼ 0

(5:107c)

L42 (cx(5=4) ) þ L43 (cy(5=4) ) þ g 14 L44 ( f5=4 ) ¼ 0

(5:107d)

O(«2 ): g14 L12 ( f1 ) g 14 ( f2 ),xx ¼ g 14 b2 [L(w2 þ W*, f0 ) þ L(w5=4 , f3=4 )] (5:108a) L21 ( f2 ) þ g 24 L22 (cx2 ) þ g24 L23 (cy2 ) þ g 24 (w2 ),xx ¼ 0

(5:108b)

L32 (cx2 ) L33 (cy2 ) þ g14 L34 ( f2 ) ¼ 0

(5:108c)

L42 (cx2 ) þ L43 (cy2 ) þ g14 L44 ( f2 ) ¼ 0

(5:108d)

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Postbuckling of Shear Deformable FGM Shells

187

(5=4)

Furthermore, we assume that w5=4 ¼ A00 , along with cx(5=4) ¼ cy(5=4) ¼ 0, (0) (1) 2 (1) 2 f1=4 ¼ f1=2 ¼ f3=4 ¼ 0, f0 ¼ B(0) 00 y =2 b00 xy, and f1 ¼ B00 y =2 b00 xy. As argued in Shen (2008c), (sin mx sin ny) is no longer the solution of Equation 5.108, hence we assume the initial buckling mode to have the form (2) (2) w2 (x, y) ¼ A(2) 00 þ A11 sin (mx ky) sin ny þ a11 cos (mx ky) cos ny

þ A(2) 02 cos 2ny

(5:109)

in which k is a continuous variable and can be determined later, and the initial geometric imperfection is assumed to have the form h i (2) (5:110) W*(x, y, «) ¼ «2 m A(2) 11 sin (mx ky) sin ny þ a11 cos (mx ky) cos ny where m is the imperfection parameter. From Equation 5.108 one has y2 (2) b(2) 00 xy þ B11 sin (mx ky) sin ny 2 (2) ¼ C(2) 11 cos (mx ky) sin ny þ c11 sin (mx ky) cos ny

f2 ¼ B(2) 00 cx2

(5:111)

(2) cy2 ¼ D(2) 11 sin (mx ky) cos ny þ d11 cos (mx ky) sin ny

and B(2) 11 ¼

g 24 m2 (2) A , g210 11

a(2) 11 ¼

(2) c(2) 11 ¼ g 14 g213 B11 ,

b2 B(0) 00 ¼

g220 (2) A , g210 11

(2) C(2) 11 ¼ g 14 g211 B11

(2) D(2) 11 ¼ g 14 g212 B11 ,

g 24 m (g210 nb þ g220 kb)

, nb g2210 g2220 (1 þ m) 2

(2) d(2) 11 ¼ g 14 g214 B11

bb(0) 00 ¼

(5:112)

3

g m g220 24

2nb g2210 g2220 (1 þ m)

where gijk are given in detail in Appendix N. ~ ~ j, y) should satisfy For the boundary layer solutions, W(x, j, y) and F(x, boundary layer equations of each order, for example,

O(«9=4 ): g 110 ~9=4 @4F @j4

~ 5=4 @4W @j4

þ g24 g 220 g 310

g120

@j3

~ x(7=4) @3C

~ 5=4 @3W @j3

~ x(7=4) @3C

@j3 g 320

þ g14 g 140

g 24 g240

~ x(7=4) @2C @j2 g430

~9=4 @4F @j4

~ 5=4 @4W @j4

þ g24

þ g 14 g220

~ y(9=4) @2C @j2

¼0

g 14

@j2

~ 5=4 @2W

~9=4 @3F @j3

~9=4 @2F

@j2 ¼0

¼ 0 (5:113a)

¼0

(5:113b) (5:113c) (5:113d)

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188

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

which leads to ~ 5=4 @4W @j4

þ 2c

~ 5=4 @2W @j2

~ 5=4 ¼ 0 þ b2 W

(5:114)

where c and b are deﬁned by Equation 5.43. The solution of Equation 5.114 can be written as ~ 5=4 ¼ a(5=4) sin fj þ a(5=4) cos fj eqj (5:115a) W 10 01 and ~9=4 ¼ b(9=4) sin fj þ b(9=4) cos fj eqj F 10 01 ~ x(7=4) ¼ c(7=4) sin fj þ c(7=4) cos fj eqj C 10 01 ~ y(9=4) ¼ 0 C

(5:115b) (5:115c) (5:115d)

where q and f are deﬁned by Equation 5.45. ^ ^ Similarly, the boundary layer solutions W(x, z, y) and F(x, z, y) can be obtained in the same manner. Then matching the regular solutions with the boundary layer solutions at ~ 5=4 ) ¼ (w5=4 þ W ~ 5=4 ),x ¼ 0 and the each end of the shell, e.g., let (w5=4 þ W ^ ^ (w5=4 þ W5=4 ) ¼ (w5=4 þ W5=4 ),x ¼ 0 at x ¼ 0 and x ¼ p edges, respectively, so that the asymptotic solutions satisfying the clamped boundary conditions are constructed as

x x x (5=4) (5=4) (5=4) (5=4) W ¼ «5=4 A00 A00 a01 cos f pﬃﬃﬃ þ a10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « « px px px (5=4) (5=4) (5=4) a01 cos f pﬃﬃﬃ þ a10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ A00 « « «

(2) (2) (2) þ «2 A(2) 00 þ A11 sin (mx ky) sin ny þ a11 cos (mx ky) cos ny þ A02 cos 2ny (2) (2) (2) A(2) 00 A11 sin ky sin ny þ a11 cos ky cos ny þ A02 cos 2ny x x x (2) m1 (2) ﬃﬃ ﬃ ﬃﬃ ﬃ ﬃﬃ ﬃ p p p cos f sin f A11 sin ky sin ny a(2) þ a exp q A(2) 01 10 00 þ (1) « « « px (2) a(2) þ (1)m a(2) 11 cos ky cos ny þ A02 cos 2ny 01 cos f pﬃﬃﬃ « h px px (2) (3) (3) þ a10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ þ «3 A00 þ A11 sin (mx ky) sin ny « « i h (3) (4) (4) 4 þ a(3) 11 cos (mx ky) cos ny þ A02 cos 2ny þ « A00 þ A11 sin (mx ky) sin ny (4) (4) (4) þ a(4) 11 cos (mx ky) cos ny þ A20 cos 2(mx ky) þ A02 cos 2ny þ A13 sin (mx ky) i (4) 5 (5:116) sin 3ny þ a(4) 13 cos (mx ky) cos 3ny þ A04 cos 4ny þ O(« )

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Postbuckling of Shear Deformable FGM Shells

189

y2 (0) y2 (1) y2 (2) b00 xyþ « B(1) b00 xy þ«2 B(2) b00 xyþB(2) 00 00 11 sin(mx ky)sin ny 2 2 2

x x x (5=4) (9=4) (9=4) þ «9=4 A00 b01 cos f pﬃﬃﬃ þb10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « «

p x (9=4) p x px y2 (3) (5=4) (9=4) þ «3 B(3) b01 cos f pﬃﬃﬃ þ b10 sin f pﬃﬃﬃ exp a pﬃﬃﬃ b00 xyþ B(3) þ A00 00 02 cos 2ny « « « 2 (2) (2) (2) þ A(2) 00 A11 sin kysin nyþa11 cos kycos nyþ A02 cos 2ny x x x (3) (2) m1 (2) b(3) A11 sin kysin ny 01 cosf pﬃﬃﬃ þb10 sin f pﬃﬃﬃ exp a pﬃﬃﬃ þ A00 þ(1) « « « p x (3) p x p x (2) þ (1)m a(2) b(3) exp a pﬃﬃﬃ 11 cos kycos nyþA02 cos 2ny 01 cos f pﬃﬃﬃ þb10 sin f pﬃﬃﬃ « « «

2 (4) y (4) (4) (4) (4) 4 þ « B00 b00 xy þB20 cos2(mx ky)þB02 cos 2nyþB13 sin(mx ky)sin 3ny 2 i (4) þb13 cos(mx ky)cos 3ny þ O(«5 ) (5:117)

F ¼ B(0) 00

x x px px (5=4) (7=4) (5=4) (7=4) Cx ¼ « A00 c10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ þ A00 c10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « « « h i (2) (2) 2 þ « C11 cos (mx ky) sin ny þ c11 sin (mx ky) cos ny h (5=2) (2) (2) (2) þ «5=2 A(2) 00 A11 sin ky sin ny þ a11 cos cos ny þ A02 cos 2ny c10 x x m1 (2) sin f pﬃﬃﬃ exp q pﬃﬃﬃ þ A(2) A11 sin ky sin ny 00 þ ( 1) « « px px (5=2) (2) þ (1)m a(2) 11 cos ky cos ny þ A02 cos 2ny c10 sin f pﬃﬃﬃ exp q pﬃﬃﬃ « « h i (3) (3) þ «3 C11 cos (mx ky) sin ny þ c11 sin (mx ky) cos ny h (4) (4) þ «4 C(4) 11 cos (mx ky) sin ny þ c11 sin (mx ky) cos ny þ C20 sin 2(mx ky) i (4) 5 (5:118) þ C(4) 13 cos (mx ky) sin 3ny þ c13 sin (mx ky) cos 3ny þ O(« ) 7=4

h i (2) Cy ¼ «2 D(2) sin (mx ky) cos ny þ d cos (mx ky) sin ny 11 11 h i (3) (3) þ «3 D(3) sin (mx ky) cos ny þ d cos (mx ky) sin ny þ D sin 2ny 11 11 02 h (4) (4) þ «4 D(4) 11 sin (mx ky) cos ny þ d11 cos (mx ky) sin ny þ D02 sin 2ny i (4) þ D(4) sin (mx ky) cos 3ny þ d cos (mx ky) sin 3ny þ O(«5 ) (5:119) 13 13 It can be seen that the solutions expressed by Equations 5.116 through 5.119 are more complicated than those of the shell subjected to axial compression. As mentioned before, all of the coefﬁcients in Equations 5.116 through 5.119 are related and can be expressed in terms of A(2) 11 , as shown (j) in Equation 5.112 for B(2) 11 , except for A00 (j ¼ 5=4, 2, 3, 4) which may be determined by substituting Equ ations 5.116 through 5.119 into closed condition, and can be written as

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells (5=4)

A00

A(4) 00

g5 1 (3) lp «1=4 [g g 5 gT1 ]T, A(2) 00 ¼ A00 ¼ 0, g24 g24 T2 2 2 1 g2 þ g2 ¼ (n2 b2 þ k2 b2 ) 210 2 220 (1 þ 2m) A(2) þ n2 b2 A(2) 11 02 8 g210 ¼ 2

(5:120)

where lp is the nondimensional compressive stress deﬁned by 2 4 (2) (2) (2) (4) lp ¼ lxs þ lxp ¼ l(0) l A « þ l A « þ p p p 11 11

(5:121)

in which lxs is the compressive stress caused by twisting. Since there is no axial load applied, the second term lxp must be zero-valued, from which k may be determined. Next, upon substitution of Equations 5.116 through 5.119 into the boundary condition (Equation 5.101c) and into Equations 5.102 and 5.103, the postbuckling equilibrium paths can be written as 2 4 (2) (4) A(2) A(2) ls ¼ l(0) s ls 11 « þ ls 11 « þ

(5:122)

2 4 (T) (2) (4) A(2) A(2) ds ¼ d(0) x dx þ dx 11 « þ dx 11 « þ

(5:123)

2 4 (4) A(2) g ¼ g(0) þ g(2) A(2) 11 « þ g 11 « þ

(5:124)

and

In Equations 5.121 through 5.124, A(2) 11 « is taken as the second perturbation parameter relating to the dimensionless maximum deﬂection. From Equation 5.126, by taking (x, y) ¼ (p=2m, p=2n), one has 2 A(2) 11 « ¼ Wm 6 Wm þ

(5:125)

where Wm is the dimensionless form of maximum deﬂection of the shell that can be written as " # 1 h W þ 5 Wm ¼ C3 ½D11 * D22 * A11 * A22 * 1=4 h

(5:126)

All symbols used in Equations 5.122 through 5.126 are also described in detail in Appendix N. Unlike the traditional perturbation scheme which only suitable for initial postbuckling analysis (Budiansky 1969, Yamaki and Kodama 1980), the present two-step perturbation technique is a powerful tool for full postbuckling analysis and can avoid the weakness of the traditional one.

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191

Equations 5.122 through 5.126 can be employed to obtain numerical results for full nonlinear postbuckling load–shortening, load–rotation, and load– deﬂection curves of FGM cylindrical shells subjected to torsion. Because of Equation 5.116, the prebuckling deformation of the shell is nonlinear. It is evident that, from Equation 5.117, there exists a shear stress along with an associate compressive stress when the shell is subjected to torsion. Such a compressive stress, no matter how small it is, will affect the buckling and postbuckling behavior of the FGM cylindrical shell as shown in Equations 5.122 through 5.124, but is missing in all the previously analyses. As a result, all the results published previously need to be reexamined. We ﬁrst examine the shear stress ts and torque MS for isotropic cylindrical shell subjected to torsion. The results are calculated and compared in Tables 5.7 and 5.8 with the theoretical and experimental results obtained by Nash (1959), Ekstrom (1963), and Suer and Harris (1959). It can be seen that the present results are in good agreement with experimental results, but lower than theoretical results of Nash (1959), Ekstrom (1963), and Suer and Harris (1959), because the compressive stresses are included in the present analysis. In addition, the postbuckling equilibrium paths for a moderately thick chromium-molybdenum steel tube (specimen J1) subjected to torsion are compared in Figure 5.11 with the experimental results of Ambrose et al. (1937). The computing data adopted here are L ¼ 47.0 in., R=h ¼ 21.283, h ¼ 0.0345 in., E ¼ 30.0 106 psi, and n ¼ 0.3. The results calculated show that when an initial geometric imperfection was present, e.g., W*=h ¼ 0.12, the present results are in reasonable agreement with the experimental results.

TABLE 5.7 Comparisons of Buckling Shear Stress ts (psi) for Isotropic Cylindrical Shells Subjected to Torsion (n ¼ 0.3) E (psi)

L (in.)

R (in.)

h (in.)

Experimental Results

Theoretical Results

28.0e þ 6

38.0

4

0.0172

6590

7493

29.0e þ 6

19.85

3

0.0075

4800

Sources Nash (1959) Present Ekstrom (1963)

6835

Present

5500 4997

TABLE 5.8 Comparisons of Buckling Load MS (in.=lb.) for Isotropic Cylindrical Shells Subjected to Torsion (n ¼ 0.3) Sources Suer and Harris (1959) Present

E (psi)

L (in.)

R (in.)

h (in.)

Experimental Results

Theoretical Results

27.0e þ 6

22.5

8.75

0.0087

9048

9448 9315

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192

Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 80 R/h = 21.283, L = 47.0 in.

ts (lb./in.2)⫻103

60

40 Present

W ∗/h = 0.0

W ∗/h = 0.12

20

Test (specimen J1) Ambrose et al. (1937)

FIGURE 5.11 Comparisons of postbuckling equilibrium paths for a chromium-molybdenum steel tube under torsion.

0

0

5 10 15 20 Shear strain e xy (in./in.) ⫻10−4

25

Tables 5.9 and 5.10 give buckling shear stress ts (in MPa) for perfect, thin (R=h ¼ 100) and moderately thick (R=h ¼ 30) Si3N4=SUS304 cylindrical shells with shell parameter Z ¼ 300 and 500 and with different values of volume fraction index N ( ¼ 0.0, 0.2, 0.5, 1.0, 2.0, and 5.0) subjected to torsion. As in TABLE 5.9 Comparisons of Buckling Shear Stress ts (MPa) for Perfect, Si3N4=SUS304 Thin Cylindrical Shells under Torsion in Thermal Environments (R=h ¼ 100, h ¼ 1 mm, T0 ¼ 300 K) Temperature Field

Z

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

300

385.331a

426.363a

458.291a

484.561a

507.646a

536.192a

b

b

b

b

b

500

394.983b

a

534.878a

T-ID (TU ¼ 300 K, TL ¼ 300 K) (TU ¼ 600 K, TL ¼ 300 K) (TU ¼ 900 K, TL ¼ 300 K) T-D (TU ¼ 600 K, TL ¼ 300 K) (TU ¼ 900 K, TL ¼ 300 K) a b

300

278.354

a

383.483

305.946

a

424.488

329.687

a

456.849

351.366

a

483.029

372.192 505.997

500 300

b

265.161 381.509a

b

293.171 422.522a

b

319.136 455.183a

b

341.134 481.770a

b

362.272 504.820a

385.870b 533.851a

500

243.525b

273.394b

301.998b

325.142b

348.629b

371.600b

300

383.482a

418.583a

446.032a

468.821a

487.868a

511.325a

500

265.161b

289.493b

310.400b

330.543b

347.712b

366.852b

300

381.509a

412.466a

435.976a

455.119a

471.683a

491.636a

500

243.525b

275.867b

290.927b

307.265b

320.771b

334.817b

Buckling mode (m, n) ¼ (1, 5). Buckling mode (m, n) ¼ (1, 3).

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TABLE 5.10 Comparisons of Buckling Shear Stress ts (MPa) for Perfect, Si3N4=SUS304 Moderately Thick Cylindrical Shells under Torsion in Thermal Environments (R=h ¼ 30, h ¼ 1 mm, T0 ¼ 300 K) Temperature Field

Z

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

(TU ¼ 300 K, TL ¼ 300 K)

300

1098.005

1214.271

1308.178

1386.625

1452.187

1537.305

500

1119.316

1240.542

1342.356

1401.286

1466.755

1554.256

(TU ¼ 600 K, TL ¼ 300 K)

300

1091.096

1206.660

1300.124

1378.409

1450.407

1534.401

500

1119.366

1238.680

1326.099

1404.326

1466.855

1550.036

(TU ¼ 900 K, TL ¼ 300 K)

300

1078.803

1194.088

1289.553

1371.918

1441.044

1527.409

500

1112.694

1230.576

1326.557

1403.768

1470.904

1556.032

(TU ¼ 600 K, TL ¼ 300 K)

300

1091.096

1186.372

1267.361

1333.170

1389.529

1460.884

500

1119.366

1218.748

1299.363

1359.732

1419.135

1482.154

(TU ¼ 900 K, TL ¼ 300 K)

300 500

1078.803 1112.694

1167.604 1198.552

1240.796 1276.269

1292.551 1329.622

1343.937 1369.840

1405.144 1435.896

T-IDa

T-Da

a

Buckling mode (m, n) ¼ (1, 2).

the case of axial compression, three thermal environmental conditions, referred to as I, II, and III, are considered. For case I, TU ¼ TL ¼ 300 K, for case II, TU ¼ 600 K, TL ¼ 300 K, and for case III TU ¼ 900 K, TL ¼ 300 K. It can be seen that, for the Si3N4=SUS304 cylindrical shell, a fully metallic shell (N ¼ 0) has lowest buckling load and that the buckling load increases as the volume fraction index N increases. The increase is about þ33% and þ38% for thin cylindrical shells and about þ34% and þ32% for moderately thick cylindrical shells, with Z ¼ 300 and 500, respectively, from N ¼ 0 to N ¼ 5 at the same thermal environmental condition II under T-D case. It is found that, for a thin cylindrical shell under condition III, the buckling load under T-ID case is lower than that under T-D case when the shell has lower volume fraction index N ¼ 0.2 (see bold numbers in Table 5.9). On the other hand, for a moderately thick shell under T-ID case, the buckling load under thermal environmental condition case II is higher than that under case I when the volume fraction index N ¼ 1.0 and 2.0, and also the buckling load under case III is higher than that under case II when N ¼ 0.5 and 5.0 (see bold numbers in Table 5.10). As a result, T-ID case may lead to an incorrect solution. It can also be found that the effect of temperature ﬁeld on the buckling shear stress under T-ID case is small, and it becomes pronounced when the temperature dependency is put into consideration. The temperature increase reduces the buckling load, and the percentage decrease is about 3% and 7% for thin cylindrical shells and about 4% and 7% for moderately thick cylindrical shells with Z ¼ 300 from thermal environmental condition case I to case III under the same volume fraction distribution N ¼ 0.2 and 2.0, respectively.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

Figure 5.12 shows the postbuckling load–shortening and load–rotation curves for Si3N4=SUS304 thin cylindrical shell with volume fraction index N ¼ 2.0 subjected to torsion under thermal environmental condition II and two cases of thermoelastic properties T-ID and T-D. It can be seen that the postbuckling equilibrium path becomes lower when the temperature800 Si3N4/SUS304 R/h = 100, Z = 300 600

N = 2.0, (m, n) = (1, 5)

ts (MPa)

TU = 600 K, TL = 300 K 400

1 2

1: T-ID 2: T-D

200

W ∗/h = 0.0 W ∗/h = 0.1

0 −0.5

0.0

(a)

0.5

∆x (mm)

1.0

1.5

800 Si3N4/SUS304 R/h = 100, Z = 300

600

N = 2.0, (m, n) = (1, 5)

ts (MPa)

TU = 600 K, TL = 300 K 1

1: T-ID

400

2

2: T-D

200

W ∗/h = 0.0 W ∗/h = 0.1

0 0.0 (b)

0.1

0.2

0.3

0.4

Γ (deg)

FIGURE 5.12 Effect of temperature dependency on the postbuckling behavior of a Si3N4=SUS304 thin cylindrical shell subjected to torsion: (a) load shortening; (b) load rotation.

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195

dependent properties are taken into account. It can also be seen that only very weak snap-through behavior of shells occurs for perfect shells. In contrast, for imperfect shells, the postbuckling behavior is stable. Figure 5.13 shows the effect of the volume fraction index N ( ¼ 0.2, 2.0, and 5.0) on the postbuckling behavior of Si3N4=SUS304 thin cylindrical shell 800 Si3N4/SUS304 600

R/h = 100, Z = 300 (m, n) = (1, 5)

ts (MPa)

TU = 600 K, TL = 300 K 400

3 2

1: N = 0.2

1

2: N = 2.0 3: N = 5.0

200

W ∗/h = 0.0 W ∗/h = 0.1

0 −0.5

0.0

(a)

0.5 ∆x (mm)

1.0

1.5

800 Si3N4/SUS304 R/h = 100, Z = 300 600

(m, n) = (1, 5)

τs (MPa)

TU = 600 K, TL = 300 K 400

1: N = 0.2

2 1

2: N = 2.0

3

3: N = 5.0 200

W ∗/h = 0.0 W ∗/h = 0.1

(b)

0 0.0

0.1

0.2 Γ (deg)

0.3

0.4

FIGURE 5.13 Effect of volume fraction index N on the postbuckling behavior of a Si3N4=SUS304 thin cylindrical shell subjected to torsion: (a) load shortening; (b) load rotation.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

subjected to torsion under thermal environmental condition II. It can be seen that the initial extension is almost the same when the volume fraction index N is greater than 2. It is found that the increase in the volume fraction index N yields an increase in postbuckling strength. Figure 5.14 gives the postbuckling load–shortening and load–rotation curves for the same cylindrical shell subjected to torsion under three thermal 800 Si3N4/SUS304 R/h = 100, Z = 300 600

N = 2.0, (m, n) = (1, 5)

ts (MPa)

TL = 300 K 1: TU = 300 K

400

2

3

1

2: TU = 600 K 3: TU = 900 K

200

W ∗/h = 0.0 W ∗/h = 0.1

0 −1.0

−0.5

0.0

(a)

0.5

1.0

1.5

2.0

∆x (mm)

800 Si3N4/SUS304 R/h = 100, Z = 300 600

N = 2.0, (m, n) = (1, 5)

ts (MPa)

TL = 300 K 400

1

1: TU = 300 K

2

2: TU = 600 K

3

3: TU = 900 K 200

W ∗/h = 0.0 W ∗/h = 0.1

0 0.0

(b)

0.1

0.2

0.3

0.4

Γ (deg)

FIGURE 5.14 Effect of temperature changes on the postbuckling behavior of a Si3N4=SUS304 thin cylindrical shell subjected to torsion: (a) load shortening; (b) load rotation.

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environmental conditions I, II, and II. It is evident that buckling loads reduce as the temperature increases, and postbuckling path becomes lower. Note that TU ¼ TL ¼ 300 K means uniform temperature ﬁeld. The initial deﬂection is not zero-valued and an initial extension occurs (curves 2 and 3 in the ﬁgure) when the heat conduction is put into consideration. Figures 5.15 through 5.17 are torsional postbuckling results for moderately thick, Si3N4=SUS304 cylindrical shells analogous to the results of Figures 5.12 2500 Si3N4/SUS304

ts (MPa)

2000

1500

1000

R/h = 30, Z = 300 N = 2.0, (m, n) = (1, 2) TU = 600 K, TL= 300 K 1

1: T-ID

2

2: T-D

W ∗/h = 0.0

500

W ∗/h = 0.1 0

2

0

(a)

4 ∆x (mm)

6

8

2500 Si3N4/SUS304 2000

R/h = 30, Z = 300 N = 2.0, (m, n) = (1, 2)

ts (MPa)

TU = 600 K, TL= 300 K 1500

1: T-ID

1

2: T-D

2

1000

W ∗/h = 0.0

500

W ∗/h = 0.1 0 0.0

(b)

0.5

1.0 Γ (deg)

1.5

2.0

FIGURE 5.15 Effect of temperature dependency on the postbuckling behavior of a Si3N4=SUS304 moderately thick cylindrical shell subjected to torsion: (a) load shortening; (b) load rotation.

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 2500 Si3N4/SUS304 R/h = 30, Z = 300

ts (MPa)

2000

(m, n) = (1, 2) TU = 600 K, TL = 300 K

1500

1: N = 0.2 2: N = 2.0

1000

1

3

2

3: N = 5.0

W ∗/h = 0.0 W ∗/h = 0.1

500

0

0

2

4 ∆x (mm)

(a)

2500

6

8

Si3N4/SUS304 R/h = 30, Z = 300

2000

(m, n) = (1, 2)

ts (MPa)

TU = 600 K, TL = 300 K 1500

1: N = 0.2

3

2: N = 2.0 3: N = 5.0

1000

1

500

0

(b)

2

W ∗/h = 0.0 W ∗/h = 0.1

0

0.5

1.0

1.5

2.0

2.5

Γ (deg)

FIGURE 5.16 Effect of volume fraction index N on the postbuckling behavior of a Si3N4=SUS304 moderately thick cylindrical shell subjected to torsion.

through 5.14. To compare these ﬁgures, it can be seen that the initial endshortening and rotation of a moderately thick cylindrical shell are larger than those of a thin cylindrical shell, and no ‘‘snap-through’’ phenomenon could be found for perfect shell. Otherwise, they lead to broadly the same conclusions as do Figures 5.12 through 5.14. In these examples, we chose the shell radius-to-thickness ratio R=h ¼ 100 and 30, Z ¼ 300, and h ¼ 1.0 mm.

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199

2500

2000

Si3N4/SUS304

1: TU = 300 K

R/h = 30, Z = 300

2: TU = 600 K

N = 2.0, (m, n) = (1, 2)

3: TU = 900 K

ts (MPa)

TL = 300 K

1500

1

2

3

1000

500

W ∗/h = 0.0 W ∗/h = 0.1

0

0

2

4 ∆x (mm)

(a)

6

8

2500

ts (MPa)

2000

1500

Si3N4/SUS304

1: TU = 300 K

R/h = 30, Z = 300

2: TU = 600 K

N = 2.0, (m, n) = (1, 2)

3: TU = 900 K

TL = 300 K 1

2

1000

3

500

W ∗/h = 0.0 W ∗/h = 0.1

0 0.0

(b)

0.5

1.0

1.5

2.0

2.5

Γ (deg)

FIGURE 5.17 Effect of temperature changes on the postbuckling behavior of a Si3N4=SUS304 moderately thick cylindrical shell subjected to torsion: (a) load shortening; (b) load rotation.

5.6

Thermal Postbuckling Behavior of FGM Cylindrical Shells

Finally, we consider thermal buckling problem for an FGM cylindrical shell. The temperature ﬁeld considered is assumed to be a uniform distribution

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Functionally Graded Materials: Nonlinear Analysis of Plates and Shells

over the shell surface and varied in the thickness direction. The material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and are assumed to be temperature-dependent. In the present case, the nonlinear differential equations in the Donnell sense are the same as in the case of axial compression, i.e., Equations 5.15 through 5.18 in ~ij() and nonlinear operator Lij() are Section 5.3, where all linear operators L deﬁned by Equation 1.37, and the forces, moments and higher order moments caused by elevated temperature are deﬁned by Equation 1.28. The two end edges of the shell are assumed to be simply supported or clamped, and to be restrained against expansion longitudinally while temperature is increased steadily, so that the boundary conditions are X ¼ 0, L: W ¼ U ¼ Cy ¼ 0,

Mx ¼ Px ¼ 0 (simply supported)

W ¼ U ¼ Cx ¼ Cy ¼ 0 (clamped)

(5:127a) (5:127b)

Also the closed (or periodicity) condition is expressed by Equation 5.20, and the average end-shortening relationship is deﬁned by Equation 5.21. Introducing dimensionless quantities of Equation 5.6, and * A22 * =D11 * D22 * ]1=4 (g T1 , gT2 ) ¼ ATx , ATy R[A11 Dx R dx ¼ , lT ¼ a0 DT (5:128) L 2[D11 * D22 * A11 * A22 * ]1=4 where a0 is an arbitrary reference value, deﬁned by Equation 3.39. Also we let "

ATx ATy

#

ðt2

DT ¼ t1

Ax Ay

DT(Z)dZ

(5:129)

where DT is the temperature rise for a uniform temperature ﬁeld, and DT ¼ TU TL for the heat conduction and the details of ATx can be found in Appendix B (for DT ¼ 0) and Appendix D (for DT 6¼ 0). T It is noted that from Equation 1.28 the thermal force N xy are zero-valued, and for a uniform temperature ﬁeld the thermal moments are also zerovalued. In contrast, for the heat conduction, the thermal moments are constants, hence the initial deﬂection of the shell with simply supported edges is not zero, but the clamped edges can prevent the transverse deﬂection from occurring. As a result, the uniform temperature ﬁeld is considered only for simply supported shells and both uniform temperature ﬁeld and heat conduction are considered for clamped shells. Then the boundary layer-type equations are in the same forms as in the case of axial compression, i.e.,

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Postbuckling of Shear Deformable FGM Shells

201

«2 L11 (W) «L12 (Cx ) «L13 (Cy ) þ «g 14 L14 (F) g 14 F,xx ¼ g14 b2 L(W þ W*, F) (5:130)

L21 (F) þ g24 L22 (Cx ) þ g 24 L23 (Cy ) «g24 L24 (W) þ g24 W,xx 1 ¼ g 24 b2 L(W þ 2W*, W) 2

(5:131)

«L31 (W) þ L32 (Cx ) L33 (Cy ) þ g14 L34 (F) ¼ 0

(5:132)

«L41 (W) L42 (Cx ) þ L43 (Cy ) þ g14 L44 (F) ¼ 0

(5:133)

The boundary conditions of Equation 5.127 become x ¼ 0, p: W ¼ dx ¼ Cy ¼ 0,

Mx ¼ Px ¼ 0

(simply supported)

W ¼ dx ¼ C x ¼ C y ¼ 0

(clamped)

(5:134a) (5:134b)

and the closed condition of Equation 5.20b becomes 2 2 @Cy @2F @Cx @2W 2@ F 2@ W þ g «g þ g g b g b g þ g b 5 24 220 522 24 240 622 @x @y @x2 @y2 @x2 @y2 0 # 1 @W 2 @W @W* g 24 b2 (5:135) þ «(gT2 g5 gT1 )lT dy ¼ 0 þg 24 W g24 b2 2 @y @y @y

2p ð

The unit end-shortening relationship becomes 1 dx ¼ 2 «1 4p g 24

2p ð ðp

g224 b2 0 0

@Cy @2F @2F @Cx þ g233 b g 5 2 þ g 24 g511 @x @y @y2 @x

2 @2W 2@ W þ g b «g24 g611 244 @x2 @y2

þ « g224 g T1 g 5 gT2 lT dx dy

1 @W 2 @W @W* g 24 g 24 2 @x @x @x (5:136)

From Equations 5.130 through 5.136, one can determine the thermal postbuckling behavior of perfect and imperfect FGM cylindrical shells under thermal loading by means of a singular perturbation technique. In the present case, the solutions are assumed to have the same forms as in the case of axial compression, i.e., Equations 5.31 through 5.33. All the necessary steps of the solution methodology are described below, but the solutions are not repeated here for convenience. First, the assumed solution form of Equation 5.31 is substituted into Equations 5.130 through 5.133 and collecting terms of the same order of «, we obtain three sets of perturbation equations for the regular and boundary layer solutions, respectively.

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Then, Equations 5.37 and 5.38 are used to solve these perturbation equations of each order step by step. At each step the amplitudes of the terms wj (x, y), fj (x, y), cxj (x, y), cyj (x, y) for the regular solutions, and of the ~ xj (x, j, y), C ~ yj (x,j,y), and W ^ j (x, z, y), F ~ j (x, j, y), F ~j (x, j, y), C ^j (x,z,y), terms W ^ ^ Cxj (x, z, y), Cyj (x,z, y) for the boundary layer solutions can be determined, respectively. By matching the regular solutions with the boundary layer solutions at each end of the shell, we obtain the asymptotic solutions satisfying the clamped boundary conditions as expressed by Equations 5.46 through 5.49. Next, upon substitution of Equations 5.46 through 5.49 into the boundary condition dx ¼ 0 and into closed condition (Equation 5.135), the thermal postbuckling equilibrium path can be written as

2 4 (0) (2) (2) (4) (2) (5:137) lT ¼ C11 lT lT A11 « þlT A11 « þ In Equation 5.137, (A(2) 11 «) is taken as the second perturbation parameter relating to the dimensionless maximum deﬂection. From Equation 5.46, by taking (x, y) ¼ (p=2m, p=2n) one has 2 A(2) 11 « ¼ Wm 7 Wm þ

(5:138a)

where Wm is the dimensionless form of the maximum deﬂection of the shell that can be written as " # 1 h W Wm ¼ þ 8 (5:138b) C3 ½D11 * D22 * A11 * A22 * 1=4 h All symbols used in Equations 5.137 and 5.138 are described in detail in Appendix O. It is noted that lT(i) (i ¼ 0, 2, . . . ) are all functions of temperature and position. Equations 5.137 and 5.138 can be employed to obtain numerical results for full thermal postbuckling load–deﬂection curves of FGM cylindrical shells. For numerical illustrations, two sets of material mixture, as shown in Section 5.3, for FGM cylindrical shells are considered. It has been shown (Shen and Li 2002) for most moderately thick cylindrical shells that the critical value of temperature rise Tcr is very high, and in such a case the thermal buckling will not occur. For this reason, in the present examples we chose the shell radius-to-thickness ratio R=h ¼ 100 and h ¼ 10 mm. The thermal buckling loads Tcr (in K) for simply supported perfect FGM cylindrical shells with different values of volume fraction index N ( ¼ 0, 0.2, 0.5, 1.0, 2.0, and 5.0) subjected to a uniform temperature rise are calculated and compared in Table 5.11. Note that, for the thermal buckling problem, it is necessary to solve Equation 5.137 by an iterative numerical procedure, as previously shown in Section 3.3. It can be seen that, for the Si3N4=SUS304 cylindrical shells, a fully metallic shell (N ¼ 0) has the lowest buckling temperature, and the buckling temperature increases as the volume fraction index

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TABLE 5.11 Comparisons of Buckling Temperatures DTcr (in K) for Simply Supported Perfect FGM Cylindrical Shells under Uniform Temperature Field with Temperature-Independent or Temperature-Dependent Properties (R=h ¼ 100, h ¼ 10 mm, T0 ¼ 300 K) Z

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

Si3N4=SUS304, T-ID 300

356.4784a

394.3238a

437.9438a

488.1880a

547.4807a

622.0067a

500

351.0783b

389.0045b

431.6156b

479.9285b

536.6419b

608.9135b

800

348.2981c

385.7825c

428.1321c

476.3186c

532.9545c

604.9021c

Si3N4=SUS304, T-D 300

298.7615a

325.5322a

355.2824a

387.9230a

424.6803a

468.5131a

500

b

b

b

b

b

460.4933b

c

800

295.1862

c

293.3418

322.1722

c

351.3019

c

382.6795

c

417.8137

320.0738

349.1119

380.5313

415.7682

458.4126c

ZrO2=Ti-6Al-4V, T-ID 300

616.5239a

519.5518a

442.9970a

385.1690a

341.8219a

310.7177a

500 800

607.6771b 602.9612c

513.0004b 508.8263c

436.9333b 433.4809c

378.8795b 376.1057c

335.2085b 332.9807c

304.3159b 302.3792c

ZrO2=Ti-6Al-4V, T-D 300

540.9151a

324.5936a

259.1421a

222.1906a

197.3951a

180.4425a

500 800

b

b

b

b

b

178.2761b 177.5914c

a b c

534.2328 530.6597c

322.0919 320.4967c

257.0812 255.8761c

220.1058 219.1405c

195.2136 194.4353c

Buckling mode (m, n) ¼ (2, 7). Buckling mode (m, n) ¼ (3, 8). Buckling mode (m, n) ¼ (4, 8).

N increases. This is expected because the metallic shell has a larger value of thermal expansion coefﬁcient af than the ceramic shell does. It is found that the increase is about þ74% under T-ID case, and about þ56% under T-D case, from N ¼ 0 to N ¼ 5. In contrast, for the ZrO2=Ti-6Al-4V cylindrical shells, the buckling temperature is decreased as the volume fraction index N increases, which conﬁrming the ﬁnding in Shen (2004). It can also be seen that the buckling temperature reduces when the temperature dependency is put into consideration. The percentage decrease is about 22% for the Si3N4=SUS304 cylindrical shell and about 42% for the ZrO2=Ti-6Al-4V cylindrical shell under the same volume fraction distribution N ¼ 2. Table 5.12 presents buckling loads Tcr (in K) for clamped perfect FGM cylindrical shells with different values of volume fraction index N ( ¼ 0, 0.2, 0.5, 1.0, 2.0, and 5.0) under heat conduction. The temperature is now set at ambient temperature 300 K on the inner surface of the shell. It is found that, for the Si3N4=SUS304 cylindrical shell, the buckling temperature under T-D case is higher than that under T-ID case when the shell has lower volume fraction index N, and the buckling temperature is no longer increased as the volume fraction index N increases.

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TABLE 5.12 Comparisons of Buckling Temperatures DTcr (in K) for CC Perfect FGM Cylindrical Shells under Heat Conduction with Temperature-Independent or TemperatureDependent Properties (R=h ¼ 100, h ¼ 10 mm, TL ¼ T0 ¼ 300 K) Z

N¼0

N ¼ 0.2

N ¼ 0.5

N ¼ 1.0

N ¼ 2.0

N ¼ 5.0

Si3N4=SUS304, T-ID 300 500

709.5367a 701.0275b

801.1807a 792.2409b

916.3505a 905.2246b

1046.498b 1035.337b

1184.8260b 1174.6604b

1320.546b 1310.380b

800

696.4399c

768.7820c

899.1711c

1028.960c

1168.1630c

1303.484c

Si3N4=SUS304, T-D 300 709.5367a

809.6627a

911.4340a

971.1096a

1049.300b

950.3516b

b

b

b

b

b

943.4778b

c

939.2562c

500 800

701.0276

c

696.0619

ZrO2=Ti-6Al-4V, T-ID 300 1227.975a 500 800

b

1213.289

c

1205.611

799.7127

c

899.4313

c

976.5723

c

1040.993

793.8150

892.8902

981.0917

711.9446a

514.2974a

406.1678b

329.1755b

240.9924b

b

b

b

b

238.7110b

c

704.7034

c

508.4977

c

401.0534

c

1036.795

325.7647

699.9917

505.2125

398.6858

324.0527

237.5206c

ZrO2=Ti-6Al-4V, T-D 300

1227.975a

392.0297a

284.2647a

230.3391b

191.8820b

145.4020b

500

b

b

b

b

b

144.3751b

c

143.8673c

800 a b c d

1213.289

c

1205.611

389.3818

c

387.8687

282.2638

c

281.1219

228.3273

c

227.4204

190.3960 189.6923

Buckling mode (m, n) ¼ (2, 7). Buckling mode (m, n) ¼ (3, 9). Buckling mode (m, n) ¼ (3, 8). Buckling mode (m, n) ¼ (4, 8).

Figure 5.18 shows the thermal postbuckling load–deﬂection curves for clamped perfect and imperfect, Si3N4=SUS304 and ZrO2=Ti-6Al-4V cylindrical shells with a volume fraction index N ¼ 0.2 under heat conduction and under two cases of thermoelastic material properties, i.e., T-ID and T-D. It can be seen that, for the ZrO2=Ti-6Al-4V shell, the thermal postbuckling equilibrium path becomes lower when the temperature-dependent properties are taken into account. It is found that, for the Si3N4=SUS304 shell, the buckling temperature and postbuckling thermal loads under T-D case are slightly higher than those under T-ID case. Figure 5.19 shows the thermal postbuckling load–deﬂection curves for the same two FGM cylindrical shells with different values of the volume fraction index N ( ¼ 0.2, 0.5, and 2.0) under heat conduction. It can be seen that the Si3N4=SUS304 shell has lower buckling temperature and postbuckling path when it has lower volume fraction index N. As argued before, this is not a general statement. For the ZrO2=Ti-6Al-4V cylindrical shell, the conclusion is reversed.

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1500 Si3N4/SUS304 R/h = 30, Z = 500 N = 0.2, (m, n) = (2, 7) TL = T0 = 300 K

1000

T-ID

∆T (K)

T-D

500 W ∗/h = 0.0

0

W ∗/h = 0.1

0.0

0.5

(a)

1.0

1.5

W ∗/h

1500

ZrO2/Ti-6Al-4V R/h = 100, Z = 300 N = 0.2, (m, n) = (2, 7) TL = T0 = 300 K

1000 ∆T (K)

TID

TD

500

W ∗/h = 0.0

0 0.0 (b)

W ∗/h = 0.1

0.5

W ∗/h

1.0

1.5

FIGURE 5.18 Effect of material properties on the thermal postbuckling behavior of FGM cylindrical shells under heat conduction: (a) (Si3N4=SUS304) shells; (b) (ZrO2=Ti-6Al-4V) shells.

Figure 5.20 compares the thermal postbuckling load–deﬂection curves for the same two FGM cylindrical shells under heat conduction and uniform temperature ﬁeld. It can be seen that the buckling temperature and postbuckling thermal loads are much lower under uniform temperature ﬁeld for the Si3N4=SUS304 shell. In contrast, for the ZrO2=Ti-6Al-4V shell, the

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Si3N4 /SUS304 3

R/h = 100, Z = 300, TL=T0 = 300 K 1: N = 0.2, (m, n = 2, 7) 2: N = 0.5, (m, n = 2, 7) 3: N = 2.0, (m, n = 3, 9)

1500

∆T (K)

2 1000 1 500 W ∗/h = 0.0

0

W ∗/h = 0.1

0.0

0.5

(a)

800

W ∗/h

1.0

1.5

ZrO2 /Ti-6Al-4V

600

R/h = 100, Z = 300, TL=T0 = 300 K 1: N = 0.2, (m, n = 2, 7) 2: N = 0.5, (m, n = 2, 7) 3: N = 2.0, (m, n = 3, 9)

∆T (K)

1 400 2 3 200 W ∗/h = 0.0

0 0.0 (b)

W ∗/h = 0.1

0.5

W ∗/h

1.0

1.5

FIGURE 5.19 Effect of volume fraction index N on the thermal postbuckling behavior of FGM cylindrical shells under heat conduction: (a) (Si3N4=SUS304) shells; (b) (ZrO2=Ti-6Al-4V) shells.

buckling temperature is slightly higher, but the postbuckling thermal loads are lower under uniform temperature ﬁeld. It is noted that in all these ﬁgures W*=h ¼ 0.05 denotes the dimensionless maximum initial geometric imperfection of the shell. Unlike solutions of

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Postbuckling of Shear Deformable FGM Shells 2500

Si3N4/SUS304 R/h = 100, Z = 300 N =2.0, TL=T0 = 300 K 1: Heat conduction, (m, n) = (3, 9) 2: Uniform temperature rise, (m, n) = (2, 7) 1

2000

∆T (K)

207

1500

W ∗/h = 0.0

1000

W ∗/h = 0.05 500

0

2

0.0

0.5

(a)

1.0

1.5

W/h

500

400

ZrO2/Ti-6Al-4V R/h = 100, Z = 300 N = 2.0, TL = T0 = 300 K 1: Heat conduction, (m, n) = (3, 9) 2: Uniform temperature rise, (m, n) = (2, 7) 1

∆T (K)

300 2 200

100

W ∗/h = 0.0 W ∗/h = 0.05

0 0.0 (b)

0.5

1.0

1.5

W/h

FIGURE 5.20 Comparisons of thermal postbuckling behavior of FGM cylindrical shells under heat conduction and uniform temperature ﬁeld: (a) (Si3N4=SUS304) shells; (b) (ZrO2=Ti-6Al-4V) shells.

Mirzavand et al. (2005) and Mirzavand and Eslami (2006), no buckling temperature could be found for imperfect FGM cylindrical shells. From Figures 5.18 through 5.20, it can be seen that the thermal postbuckling equilibrium path is stable or weakly unstable and the shell structure is virtually imperfection-insensitive for both T-ID and T-D cases.

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Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A001 Final Proof page 211 4.12.2008 11:09am Compositor Name: VAmoudavally

Appendix A In Equations 2.14 and 2.16 * , (F12 * þ F21 * þ 4F66 * )=2, F22 * ]=D11 *, (g110 , g112 , g114 ) ¼ (4=3h2 )[F11 * 4F11 * =3h2 , (D12 * þ 2D66 * ) 4(F12 * þ 2F66 * )=3h2 ]=D11 * (g 120 , g122 ) ¼ [D11 2 2 * þ 2D66 * ) 4(F21 * þ 2F66 * )=3h , D22 * 4F22 * =3h =D11 * (g131 , g133 ) ¼ (D12 * , (B11 * þ B22 * 2B66 * ), B12 * ]=[D11 * D22 * A11 * A22 * ]1=4 (g140 , g142 , g144 ) ¼ [B21 * þ A66 * =2, A11 * )=A22 * (g212 , g214 ) ¼ (A12 * 4E21 * =3h2 , (B11 * B66 * ) 4(E11 * -E66 * )=3h2 =[D11 * D22 * A11 * A22 * ]1=4 (g220 , g222 ) ¼ B21 * B66 * ) 4(E22 * E66 * )=3h2 , B12 * 4E12 * =3h2 =[D11 * D22 * A11 * A22 * ]1=4 (g231 , g233 ) ¼ (B22 * , (E11 * þ E22 * 2E66 * ), E12 * ]=[D11 * D22 * A11 * A22 * ]1=4 (g240 , g242 , g244 ) ¼ (4=3h2 )[½E21 *, (g31 , g41 ) ¼ (a2 =p2 ) A55 8D55 =h2 þ 16F55 =h4 , A44 8D44 =h2 þ 16F44 =h4 =D11 2 2 2 * * * * * * * (g310 , g312 ) ¼ (4=3h ) F11 4H11 =3h ,(F21 þ 2F66 ) 4(H12 þ 2H66 )=3h ]=D11 * 8F11 * =3h2 þ 16H11 * =9h4 , D66 * 8F66 * =3h2 þ 16H66 * =9h4 =D11 * (g320 , g322 ) ¼ D11 2 4 * þ D66 * ) 4(F12 * þ F21 * þ 2F66 * )=3h þ 16(H12 * þ H66 * )=9h =D11 * g331 ¼ (D12 2 2 2 * þ 2F66 * ) 4(H12 * þ 2H66 * )=3h , F22 * 4H22 * =3h =D11 * (g411 , g413 ) ¼ (4=3h ) (F12 2 4 2 4 * * * * * * * (g430 , g432 ) ¼ D66 8F66 =3h þ 16H66 =9h , D22 8F22 =3h þ 16H22 =9h =D11 1=4 * 4E11 * =3h2 , B22 * 4E22 * =3h2 =[D11 * D22 * A11 * A22 *] (g511 , g522 ) ¼ B11 * , E22 * )=½D11 * D22 * A11 * A22 * 1=4 (g611 , g622 ) ¼ (4=3h2 )(E11 * , B22 * )=[D11 * D22 * A11 * A22 * ]1=4 (g711 , g722 ) ¼ (B11

* 4F12 * =3h2 , D12 * 4F21 * =3h2 =D11 * (g812 , g821 ) ¼ D12 * , F21 * )=D11 * (g912 , g921 ) ¼ (4=3h2 )(F12

(A:1)

211

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A001 Final Proof page 212 4.12.2008 11:09am Compositor Name: VAmoudavally

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A002 Final Proof page 213 4.12.2008 11:10am Compositor Name: VAmoudavally

Appendix B In Equation 2.9 h 1 1 (am ac )(Em Ec ) þ [ac (Em Ec ) þ (am ac )Ec ] þ ac Ec 1n 2N þ 1 Nþ1 h2 N T Dx ¼ (am ac )(Em Ec ) 1n 2(N þ 1)(2N þ 1) N þ[ac (Em Ec ) þ (am ac )Ec ] 2(N þ 1)(N þ 2) 4 h 1 3 FTx ¼ (am ac )(Em Ec ) 1n 8(2N þ 1) 8(N þ 1)(2N þ 1) 3 3 þ 2(N þ 1)(2N þ 1)(2N þ 3) 2(N þ 1)(N þ 2)(2N þ 1)(2N þ 3) 1 3 þ ½ac (Em Ec ) þ (am ac )Ec 8(N þ 1) 4(N þ 1)(N þ 2) 3 6 (B:1) þ (N þ 1)(N þ 2)(N þ 3) (N þ 1)(N þ 2)(N þ 3)(N þ 4) ATx ¼

213

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A002 Final Proof page 214 4.12.2008 11:10am Compositor Name: VAmoudavally

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A003 Final Proof page 215 4.12.2008 10:40am Compositor Name: VAmoudavally

Appendix C In Equation 2.19 "

M(1) x

S(1) x

M(1) y

S(1) y

# ¼

g T3 16 DT 2 p (W=h) g T4

(g T3 g T6 ) (g T4 g T7 )

A(1) 11

(C:1)

and in Equations 2.32 through 2.34 4 2 2 2 * A(0) W ¼ p DT (g T3 m þ g T4 n b ) Q11 4 A(1) W ¼ p C11 [S11 CW1 ] 4 A(2) W ¼ p QW22

A(3) W

¼ p C11 4

h 1=4

* D22 * A11 * A22 * ½D11

1 (a313 þ a331 )[S11 CW1 ] þ g 14 g24 Q2 16

h2 * D22 * A11 * A22 * ]1=2 [D11

* A(0) MX ¼ 92416DT(g T3 QX11 )=11025 þ CX0 2 A(1) MX ¼ p QX11 2 A(2) MX ¼ p QX22

h * * * A22 * ]1=4 [D11 D22 A11

2 A(3) MX ¼ p [QX11 (a313 þ a331 ) QX33 ]

h2 * D22 * A11 * A22 * ]1=2 [D11

* A(0) MY ¼ 92416DT(g T4 QY11 )=11025 þ CY0 2 A(1) MY ¼ p QY11 2 A(2) MY ¼ p QY22

h * * * A22 * ]1=4 [D11 D22 A11

2 A(3) MY ¼ p [QY11 (a313 þ a331 ) QY33 ]

h2 * D22 * A11 * A22 * ]1=2 [D11

(C:2)

in which Q2 ¼

m4 n4 b4 þ þ C22 , g7 g6

S11 ¼ g08 þ g 14 g24 m2 n2 b2

S13 ¼ g138 þ g 14 g24 9m2 n2 b2

g135 g137 , g136

g05 g07 , g06

S31 ¼ g318 þ g 14 g24 9m2 n2 b2

g315 g317 g316

215

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A003 Final Proof page 216 4.12.2008 10:40am Compositor Name: VAmoudavally

216 QW22

Appendix C 1 g9 g05 2 2 2 g8 , ¼ g g m nb þ þ4 g6 g7 g06 3C11 14 24

* ¼ g08 * þ g 14 g24 Q11

g6 ¼ 1 þ g14 g 24 g2230

4m2 , g 41 þ g 322 4m2

g7 ¼ g224 þ g 14 g24 g 2223

g8 ¼ g140 g120 g220

4m2 , g 31 þ g320 4m2

g9 ¼ g 144 g 133 g 233

g18 ¼ g722 g812 g220

4m2 , g 31 þ g320 4m2

g19 ¼ g711 g821 g233

J13 ¼ S13 C13 , a313 ¼

* g07 g05 , g06

4n2 b2 , g31 þ g322 4n2 b2

4n2 b2 , g41 þ g432 4n2 b2 4n2 b2 , g 41 þ g432 4n2 b2

J31 ¼ S31 C31 ,

1 m2 g 14 g24 , g 7 J13 16

a331 ¼

1 n2 b2 g14 g 24 , g 6 J31 16

g05 QX11 ¼ (g110 m2 þ g921 n2 b2 ) þ g14 g 24 (g140 m2 þ g711 n2 b2 ) g06 g g g g g01 g05 04 02 05 03 þ g 821 n2 b2 þ g120 m2 g 14 g24 g14 g 24 g00 g00 g06 g00 g00 g06 2 2 2 1 m g19 n b g 8 1 þ þ CX2 , QX33 ¼ g14 g 24 (QX13 þ QX31 ) QX22 ¼ g 14 g24 g7 g6 8 16 m2 g120 g132 þ 9n2 b2 g821 g131 g135 m4

2 2 2 g140 m þ g711 9n b QX13 ¼ g14 g 24 g 7 J13 g130 g136 2 4 2 2

m g120 g134 þ 9n b g821 g133 m þ g110 m2 þ g921 9n2 b2 þ g7 J13 g130 2 2 2 n4 b4

9m g 120 g312 þ n b g 821 g311 g315 QX31 ¼ g14 g 24 g140 9m2 þ g711 n2 b2 g 6 J31 g310 g316 4 4

2 2 2 nb 9m g120 g314 þ n b g821 g313 þ g110 9m2 þ g921 n2 b2 þ g6 J31 g310

g05 QY11 ¼ g912 m2 þ g114 n2 b2 þ g14 g 24 g722 m2 þ g144 n2 b2 g06 g02 g05 g01 g05 2 g04 2 2 g03 þ g 133 n b þ g812 m g 14 g24 g14 g 24 g00 g00 g06 g00 g00 g06 2 2 2 1 m g9 n b g18 1 þ þ CY2 , QY33 ¼ g 14 g24 (QY13 þ QY31 ) QY22 ¼ g 14 g24 g7 g6 8 16 2 2 2 m4

m g g 812 132 þ 9n b g 133 g131 g135 g722 m2 þ g144 9n2 b2 QY13 ¼ g14 g 24 g 7 J13 g130 g136 4 2 2 2

m m g812 g134 þ 9n b g133 g133 þ g912 m2 þ g114 9n2 b2 þ g7 J13 g130

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A003 Final Proof page 217 4.12.2008 10:40am Compositor Name: VAmoudavally

Appendix C

217

n4 b4 9m2 g812 g312 þ n2 b2 g133 g311 g315 (g 722 9m2 þ g 144 n2 b2 ) g6 J31 g310 g316 2 2 2 n4 b4 9m g g þ n b g g 812 314 133 313 þ (g 912 9m2 þ g114 n2 b2 ) þ g6 J31 g310 g* g* g* g* * ¼ g120 m2 04 þ g821 n2 b2 03 , QY11 * ¼ g812 m2 04 þ g133 n2 b2 03 QX11 g00 g00 g00 g00 QY31 ¼ g14 g24

g00 ¼ (g31 þ g320 m2 þ g322 n2 b2 )(g41 þ g430 m2 þ g432 n2 b2 ) g2331 m2 n2 b2 g01 ¼ (g31 þ g320 m2 þ g322 n2 b2 )(g231 m2 þ g233 n2 b2 ) g331 n2 b2 (g 220 m2 þ g222 n2 b2 ) g02 ¼ (g41 þ g430 m2 þ g432 n2 b2 )(g220 m2 þ g222 n2 b2 ) g331 m2 (g 231 m2 þ g233 n2 b2 ) g03 ¼ (g31 þ g320 m2 þ g322 n2 b2 )(g41 g411 m2 g413 n2 b2 ) g331 m2 (g 31 g310 m2 g312 n2 b2 ) * ¼ (g31 þ g320 m2 þ g322 n2 b2 )(gT4 gT7 ) g331 m2 (g T3 gT6 ) g03 g04 ¼ (g41 þ g430 m2 þ g432 n2 b2 )(g31 g310 m2 g312 n2 b2 ) g331 n2 b2 (g41 g411 m2 g 413 n2 b2 ) * ¼ (g41 þ g430 m2 þ g432 n2 b2 )(gT3 gT6 ) g331 n2 b2 (g T4 gT7 ) g04 g05 ¼ (g240 m4 þ g 242 m2 n2 b þ g244 n4 b4 ) m2 (g 220 m2 þ g222 n2 b2 )g04 þ n2 b2 (g 231 m2 þ g 233 n2 b2 )g03 , g00 * þ n2 b2 (g 231 m2 þ g233 n2 b2 )g03 * m2 (g 220 m2 þ g 222 n2 b2 )g04 * ¼ , g05 g00 þ

g06 ¼ (m4 þ 2g212 m2 n2 b2 þ g 214 n4 b4 ) þ g14 g24

m2 (g 220 m2 þ g222 n2 b2 )g02 þ n2 b2 (g 231 m2 þ g233 n2 b2 )g01 g00

g07 ¼ (g140 m4 þ g 142 m2 n2 b þ g144 n4 b4 )

m2 (g 120 m2 þ g122 n2 b2 )g02 þ n2 b2 (g 131 m2 þ g 133 n2 b2 )g01 g00

g08 ¼ (g110 m4 þ 2g 112 m2 n2 b2 þ g114 n4 b4 ) m2 (g 120 m2 þ g122 n2 b2 )g04 þ n2 b2 (g 131 m2 þ g 133 n2 b2 )g03 g00 2 2 2 2 * * m (g 120 m þ g 122 n b )g04 þ n2 b2 (g 131 m2 þ g133 n2 b2 )g03 * ¼ g08 g00 þ

g130 ¼ (g31 þ g320 m2 þ g322 9n2 b2 )(g41 þ g430 m2 þ g432 9n2 b2 ) g2331 9m2 n2 b2 g131 ¼ (g31 þ g320 m2 þ g322 9n2 b2 )(g231 m2 þ g233 9n2 b2 ) g331 9n2 b2 (g 220 m2 þ g222 n2 b2 )

g132 ¼ g 41 þ g430 m2 þ g432 9n2 b2 g220 m2 þ g 222 9n2 b2 g331 m2 g231 m2 þ g233 9n2 b2

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A003 Final Proof page 218 4.12.2008 10:40am Compositor Name: VAmoudavally

218

Appendix C

g133 ¼ g31 þ g320 m2 þ g322 9n2 b2 g41 g 411 m2 g413 9n2 b2

g331 m2 g31 g310 m2 g312 9n2 b2

g134 ¼ g41 þ g430 m2 þ g432 9n2 b2 g31 g 310 m2 g312 9n2 b2

g331 9n2 b2 g41 g 411 m2 g413 9n2 b2

g135 ¼ g240 m4 þ 9g242 m2 n2 b þ 81g 244 n4 b4

m2 g220 m2 þ g222 9n2 b2 g134 þ 9n2 b2 g231 m2 þ g 233 9n2 b2 g133 , þ g130

4 g136 ¼ m þ 18g212 m2 n2 b2 þ 81g214 n4 b4

m2 g 220 m2 þ g222 9n2 b2 g132 þ 9n2 b2 g231 m2 þ g233 9n2 b2 g131 þ g14 g24 g130

g137 ¼ g140 m4 þ 9g142 m2 n2 b þ 81g 144 n4 b4

m2 g120 m2 þ g122 9n2 b2 g132 þ 9n2 b2 g131 m2 þ g 133 9n2 b2 g131 g130

4 2 2 2 g138 ¼ g110 m þ 18g112 m n b þ g 114 81n4 b4

m2 g120 m2 þ g122 9n2 b2 g134 þ 9n2 b2 g131 m2 þ g 133 9n2 b2 g133 þ g130

g310 ¼ g31 þ g320 9m2 þ g322 n2 b2 g41 þ g 430 9m2 þ g432 n2 b2 g2331 9m2 n2 b2

g311 ¼ g31 þ g320 9m2 þ g322 n2 b2 g231 9m2 þ g233 n2 b2 g331 n2 b2 g220 9m2 þ g222 n2 b2

g312 ¼ g41 þ g430 9m2 þ g432 n2 b2 g220 9m2 þ g222 n2 b2 g331 9m2 g231 9m2 þ g233 n2 b2

g313 ¼ g31 þ g320 9m2 þ g322 n2 b2 g41 g 411 9m2 g413 n2 b2

g331 9m2 g31 g310 9m2 g312 n2 b2

g314 ¼ g41 þ g430 9m2 þ g432 n2 b2 g31 g 310 9m2 g312 n2 b2

g331 n2 b2 g41 g411 9m2 g413 n2 b2

g315 ¼ 81g 240 m4 þ 9g242 m2 n2 b þ g 244 n4 b4

9m2 g220 m2 þ g222 n2 b2 g314 þ n2 b2 g231 9m2 þ g 233 n2 b2 g313 , þ g310 g316 ¼ (81m4 þ 18g212 m2 n2 b2 þ g214 n4 b4 ) þ g14 g24

9m2 (g 220 9m2 þ g222 n2 b2 )g312 þ n2 b2 (g 231 9m2 þ g233 n2 b2 )g311 , g310

g317 ¼ (81g140 m4 þ 9g142 m2 n2 b þ g144 n4 b4 )

9m2 (g 120 9m2 þ g 122 n2 b2 )g312 þ n2 b2 (g 131 9m2 þ g133 n2 b2 )g311 , g310

g318 ¼ (81g110 m4 þ 18g112 m2 n2 b2 þ g114 n4 b4 ) þ

9m2 (g 120 9m2 þ g 122 n2 b2 )g314 þ n2 b2 (g 131 9m2 þ g133 n2 b2 )g313 g310

(C:3)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A003 Final Proof page 219 4.12.2008 10:40am Compositor Name: VAmoudavally

Appendix C

219

and C11 ¼

p2 mn 16

C11 ¼ 1

(for uniform load)

(for sinusoidal load)

(C:4a) (C:4b)

in the above equations, for the case of movable edges CW1 ¼ CX0 ¼ CY0 ¼ CX2 ¼ CY2 ¼ C22 ¼ C13 ¼ C31 ¼ 0

(C:5a)

and for the case of immovable edges

CW1 ¼ g 14 gT1 m2 þ gT2 n2 b2 DT, * D22 * A11 * A22 * ]1=4 [D11 h * * * A22 * ]1=4 [D D A 11 22 11 ¼ p2 g14 (g 144 g T1 þ g722 gT2 )DT h

4 2 4 4 2 2 2 m þ g 24 n b þ 2g5 m n b ¼2 g224 g25

¼ g 14 gT1 m2 þ 9gT2 n2 b2 DT, C31 ¼ g14 (9g T1 m2 þ gT2 n2 b2 )DT

CX0 ¼ p2 g14 (g 711 g T1 þ g140 gT2 )DT CY0 C22 C13

CX2 ¼ g 711

m2 þ g5 n2 b2 g5 m2 þ g224 n2 b2 þ g , 140 g224 g25 g 224 g 25

CY2 ¼ g 144

m2 þ g5 n2 b2 g m2 þ g2 n2 b2 þ g 722 5 2 24 2 2 2 g24 g5 g 24 g 5

(C:5b)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A003 Final Proof page 220 4.12.2008 10:40am Compositor Name: VAmoudavally

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A004 Final Proof page 221

6.12.2008 3:44pm Compositor Name: DeShanthi

Appendix D In Equation 2.44, when DT 6¼ 0 ATx ¼

h TU T0 1 (am ac )(Em Ec ) 1 nf DT 2N þ 1 1 þ ac Ec þ ½ac (Em Ec ) þ Ec (am ac ) Nþ1 h TL TU 1 kmc (am ac )(Em Ec ) þ 1 nf CDT 2N þ 2 (N þ 1)(3N þ 2)kc k2mc k3mc k4mc þ 2 3 (2N þ 1)(4N þ 2)kc (3N þ 1)(5N þ 2)kc (4N þ 1)(6N þ 2)k4c k5mc 1 þ ½ac (Em Ec ) þ Ec (am ac ) Nþ2 (5N þ 1)(7N þ 2)k5c þ

kmc k2mc k3mc þ (N þ 1)(2N þ 2)kc (2N þ 1)(3N þ 2)k2c (3N þ 1)(4N þ 2)k3c k4mc k5mc þ (4N þ 1)(5N þ 2)k4c (5N þ 1)(6N þ 2)k5c 1 kmc k2mc þ ac E c þ 2 (N þ 1)(N þ 2)kc (2N þ 1)(2N þ 2)k2c

k3mc k4mc k5mc þ 3 4 (3N þ 1)(3N þ 2)kc (4N þ 1)(4N þ 2)kc (5N þ 1)(5N þ 2)k5c

(D:1)

221

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A004 Final Proof page 222

222 DTx

6.12.2008 3:44pm Compositor Name: DeShanthi

Appendix D h2 TU T0 1 1 (am ac )(Em Ec ) ¼ 1 nf DT 2(2N þ 1) (2N þ 1)(2N þ 2) 1 1 þ ½ac (Em Ec ) þ Ec (am ac ) 2(N þ 1) (N þ 1)(N þ 2) h2 TL TU 1 1 þ (am ac )(Em Ec ) 1 nf CDT 2(2N þ 2) (2N þ 2)(2N þ 3) kmc 1 1 (N þ 1)kc 2(3N þ 2) (3N þ 2)(3N þ 3) k2mc 1 1 þ (2N þ 1)k2c 2(4N þ 2) (4N þ 2)(4N þ 3) k3mc 1 1 (3N þ 1)k3c 2(5N þ 2) (5N þ 2)(5N þ 3) k4mc 1 1 þ (4N þ 1)k4c 2(6N þ 2) (6N þ 2)(6N þ 3) k5mc 1 1 (5N þ 1)k5c 2(7N þ 2) (7N þ 2)(7N þ 3) 1 1 þ ½ac (Em Ec ) þ Ec (am ac ) 2(N þ 2) (N þ 2)(N þ 3) kmc 1 1 (N þ 1)kc 2(2N þ 2) (2N þ 2)(2N þ 3) k2mc 1 1 þ (2N þ 1)k2c 2(3N þ 2) (3N þ 2)(3N þ 3) k3mc 1 1 (3N þ 1)k3c 2(4N þ 2) (4N þ 2)(4N þ 3) k4mc 1 1 þ (4N þ 1)k4c 2(5N þ 2) (5N þ 2)(5N þ 3) k5mc 1 1 (5N þ 1)k5c 2(6N þ 2) (6N þ 2)(6N þ 3) 1 kmc 1 1 þ a c Ec 12 (N þ 1)kc 2(N þ 2) (N þ 2)(N þ 3) 2 kmc 1 1 þ (2N þ 1)k2c 2(2N þ 2) (2N þ 2)(2N þ 3) k3mc 1 1 (3N þ 1)k3c 2(3N þ 2) (3N þ 2)(3N þ 3) k4mc 1 1 þ (4N þ 1)k4c 2(4N þ 2) (4N þ 2)(4N þ 3) k5mc 1 1 (D:2) (5N þ 1)k5c 2(5N þ 2) (5N þ 2)(5N þ 3)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A004 Final Proof page 223

Appendix D FTx

6.12.2008 3:44pm Compositor Name: DeShanthi

223

h4 TU T0 1 3 (am ac )(Em Ec ) ¼ 1 nf DT 8(2N þ 1) 4(2N þ 1)(2N þ 2) 3 6 þ (2N þ 1)(2N þ 2)(2N þ 3) (2N þ 1)(2N þ 2)(2N þ 3)(2N þ 4) 1 3 þ ½ac (Em Ec ) þ Ec (am ac ) 8(N þ 1) 4(N þ 1)(N þ 2) 3 6 þ (N þ 1)(N þ 2)(N þ 3) (N þ 1)(N þ 2)(N þ 3)(N þ 4) h4 TL TU 1 (am ac )(Em Ec ) þ 1 nf CDT 8(2N þ 2) 3 3 þ 4(2N þ 2)(2N þ 3) (2N þ 2)(2N þ 3)(2N þ 4) 6 kmc 1 (N þ 1)kc 8(3N þ 2) (2N þ 2)(2N þ 3)(2N þ 4)(2N þ 5) 3 3 þ 4(3N þ 2)(3N þ 3) (3N þ 2)(3N þ 3)(3N þ 4) 6 k2mc þ (2N þ 1)k2c (3N þ 2)(3N þ 3)(3N þ 4)(3N þ 5) 1 3 3 þ 8(4N þ 2) 4(4N þ 2)(4N þ 3) (4N þ 2)(4N þ 3)(4N þ 4) 6 k3mc 1 3 (3N þ 1)kc 8(5N þ 2) (4N þ 2)(4N þ 3)(4N þ 4)(4N þ 5) 3 3 þ 4(5N þ 2)(5N þ 3) (5N þ 2)(5N þ 3)(5N þ 4) 6 k4mc 1 þ 4 (4N þ 1)kc 8(6N þ 2) (5N þ 2)(5N þ 3)(5N þ 4)(5N þ 5) 3 3 þ 4(6N þ 2)(6N þ 3) (6N þ 2)(6N þ 3)(6N þ 4) 6 k5mc 1 5 (5N þ 1)kc 8(7N þ 2) (6N þ 2)(6N þ 3)(6N þ 4)(6N þ 5) 3 3 þ 4(7N þ 2)(7N þ 3) (7N þ 2)(7N þ 3)(7N þ 4) 6 þ ½ac (Em Ec ) þ Ec (am ac ) (7N þ 2)(7N þ 3)(7N þ 4)(7N þ 5)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A004 Final Proof page 224

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Appendix D

1 3 3 6 þ 8(N þ 2) 4(N þ 2)(N þ 3) (N þ 2)(N þ 3)(N þ 4) (N þ 2)(N þ 3)(N þ 4)(N þ 5) kmc 1 3 3 þ (N þ 1)kc 8(2N þ 2) 4(2N þ 2)(2N þ 3) (2N þ 2)(2N þ 3)(2N þ 4) 6 k2mc 1 3 þ (2N þ 1)k2c 8(3N þ 2) 4(3N þ 2)(3N þ 3) (2N þ 2)(2N þ 3)(2N þ 4)(2N þ 5) 3 6 þ (3N þ 2)(3N þ 3)(3N þ 4) (3N þ 2)(3N þ 3)(3N þ 4)(3N þ 5) k3mc 1 3 3 þ (3N þ 1)k3c 8(4N þ 2) 4(4N þ 2)(4N þ 3) (4N þ 2)(4N þ 3)(4N þ 4) 6 k4mc 1 3 þ (4N þ 1)k4c 8(5N þ 2) 4(5N þ 2)(5N þ 3) (4N þ 2)(4N þ 3)(4N þ 4)(4N þ 5) 3 6 þ (5N þ 2)(5N þ 3)(5N þ 4) (5N þ 2)(5N þ 3)(5N þ 4)(5N þ 5) k5mc 1 3 3 þ (5N þ 1)k5c 8(6N þ 2) 4(6N þ 2)(6N þ 3) (6N þ 2)(6N þ 3)(6N þ 4) 6 1 kmc 1 þ ac Ec (6N þ 2)(6N þ 3)(6N þ 4)(6N þ 5) 80 (N þ 1)kc 8(N þ 2) 3 3 6 þ 4(N þ 2)(N þ 3) (N þ 2)(N þ 3)(N þ 4) (N þ 2)(N þ 3)(N þ 4)(N þ 5) k2mc 1 3 3 þ þ (2N þ 1)k2c 8(2N þ 2) 4(2N þ 2)(2N þ 3) (2N þ 2)(2N þ 3)(2N þ 4) 6 k3mc 1 3 (3N þ 1)k3c 8(3N þ 2) 4(3N þ 2)(3N þ 3) (2N þ 2)(2N þ 3)(2N þ 4)(2N þ 5) 3 6 þ (3N þ 2)(3N þ 3)(3N þ 4) (3N þ 2)(3N þ 3)(3N þ 4)(3N þ 5) k4mc 1 3 3 þ þ 4 (4N þ 1)kc 8(4N þ 2) 4(4N þ 2)(4N þ 3) (4N þ 2)(4N þ 3)(4N þ 4) 6 k5mc 1 (5N þ 1)k5c 8(5N þ 2) (4N þ 2)(4N þ 3)(4N þ 4)(4N þ 5) 3 3 þ 4(5N þ 2)(5N þ 3) (5N þ 2)(5N þ 3)(5N þ 4) 6 (D:3) (5N þ 2)(5N þ 3)(5N þ 4)(5N þ 5)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A005 Final Proof page 225

6.12.2008 3:46pm Compositor Name: DeShanthi

Appendix E In Equation 2.45 "

M0x

S0x

M0y

S0y

#

" 16 g T3 ¼ 2 p g T4

(g T3 g T6 ) (g T4 g T7 )

# DT

h * D22 * A11 * A22 * ]1=4 [D11

(E:1)

and in Equations 2.46 through 2.49 * D22 * A11 * A22 * ]1=4 [D11 h 4 * D22 * A11 * A22 * ]1=4 1 m n4 b4 [D11 ¼ g 14 g24 þ J13 g 7 J31 g 6 h 16 5,776 1 (gT3 g T6 )m2 g104 þ (gT4 gT7 )n2 b2 g103 ¼ CX0 16DT g g00 11,025 T3 mn

A(1) W ¼ A(3) W A(0) MX

* D22 * A11 * A22 * ]1=4 [D11 h * * * A22 * ]1=4 D [D 11 22 A11 ¼ p2 QX22 h * * * A22 * ]1=4 D A [D 11 22 11 ¼ p2 QX33 h

2 A(1) MX ¼ p QX11

A(2) MX A(3) MX

* D22 * A11 * A22 * ]1=4 [D11 2 A(0) ¼ p g (g g þ g g )DT 14 144 T1 722 T2 MY h 5,776 1 (g T3 gT6 )m2 g204 þ (g T4 g T7 )n2 b2 g203 gT4 16DT g00 11,025 mn * D22 * A11 * A22 * ]1=4 [D11 h * * * A22 * ]1=4 D A11 [D 11 22 ¼ p2 QY22 h * D22 * A11 * A22 * ]1=4 [D11 ¼ p2 QY33 h

2 A(1) MY ¼ p QY11

A(2) MY A(3) MY

225

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A005 Final Proof page 226

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6.12.2008 3:46pm Compositor Name: DeShanthi

Appendix E

(gT3 gT6 )m2 g102 þ (gT4 gT7 )n2 b2 g101 16 2 2 2 l¼ 2 DT gT3 m þ gT4 n b g00 p mnG08 h * D22 * A11 * A22 * ]1=4 [D11 4 g g g05 Q2 ¼ 2 g14 g24 m2 n2 b2 8 þ 9 þ 4 g6 g7 g06 3p G08 4 4 4 1 m n b g14 g 24 þ þ C33 Q3 ¼ 2Q22 (E:2) g7 g6 16G08

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A006 Final Proof page 227 4.12.2008 11:11am Compositor Name: VAmoudavally

Appendix F In Equations 3.33 and 3.34 (2) (4) l(0) ¼ x , lx , l x d(2) x d(4) x

1 P (S0 , S2 , S4 ), d(0) x ¼ C00 lx dx , 4b g 14 C11 1 ¼ C11 (1 þ 2m), 32b2 1 m4 n4 b4 2 (1 þ m)2 (1 þ 2m)2 ¼ g14 g 24 C11 þ J31 J13 g224 256b2 2

(F:1)

in which (with g08, g138, g318, etc. are deﬁned as in Appendix C) Q11 1 SP0 , S2 ¼ g 14 g24 Q2 (1 þ 2m), (1 þ m) 16 1 2 2 S4 ¼ g g C11 (C24 C44 ), 256 14 24 Q11 ¼ g08 , Q13 ¼ g138 , Q31 ¼ g318 4 m m4 n4 b4 2 2 4 4 , Q2 ¼ þ n b þ C22 , C24 ¼ 2(1 þ m) (1 þ 2m) Q2 þ J31 g224 J13 g 224 m8 n8 b8 þ C44 ¼ (1 þ m)(1 þ 2m)[2(1 þ m)2 þ (1 þ 2m)] J31 J13 g 424 S0 ¼

J13 ¼ Q13 C11 (1 þ m) Q11 C13 þ J P , J31 ¼ Q31 C11 (1 þ m) Q11 C31 J P (F:2) in the above equations, for the case of four edges movable C00 ¼ g 24 , C11 ¼ C13 ¼ m2 , C31 ¼ 9m2 , C22 ¼ 0, SP0 ¼ J P ¼ 0, 1 2 g 24 gT1 g5 gT2 DT þ g224 g P1 g5 gP2 DV dPx ¼ 2 4b g24

(F:3)

227

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A006 Final Proof page 228 4.12.2008 11:11am Compositor Name: VAmoudavally

228

Appendix F

and for the case of unloaded edges immovable C00 ¼

1 g 2 g25 g24 24

C11 ¼ m2 þ g 5 n2 b2 , C13 ¼ m2 þ 9g 5 n2 b2 , C31 ¼ 9m2 þ g5 n2 b2 , C22 ¼ 2n4 b4 , SP0 ¼ g14 n2 b2 [(g T2 g5 gT1 )DT þ (gP2 g 5 gP1 )DV], C00 dPx ¼ 2 (g T1 DT þ gP1 DV), 4b J P ¼ 8g14 m2 n2 b2 (1 þ m)[(gT2 g5 gT1 )DT þ (gP2 g 5 gP1 )DV]:

(F:4)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A007 Final Proof page 229 4.12.2008 11:11am Compositor Name: VAmoudavally

Appendix G In Equation 3.53 (2) (4) (l(0) T , l T , lT ) ¼

1 (S0 , S2 , S4 ) g 14 C11

(G:1)

in which [with g08, g138, g318, etc. are deﬁned as in Appendix C] S0 ¼

Q11 SP0 , (1 þ m)

S2 ¼

1 g 14 Q2 (1 þ 2m), 16 g 24

S4 ¼

1 g 214 C11 (C24 C44 ) 256 g 224

SP0 ¼ g 14 n2 b2 (gP2 g5 gP1 )DV, Q11 ¼ g08 , Q13 ¼ g138 , Q31 ¼ g318 3g224 g 25 m4 þ g224 n4 b4 þ 4g 5 g 224 m2 n2 b2 Q2 ¼ g224 g 25 4 m g 224 n4 b4 2 2 C24 ¼ 2(1 þ m) (1 þ 2m) Q2 þ J13 J31 8 m g 424 n8 b8 2 þ C44 ¼ (1 þ m)(1 þ 2m)[2(1 þ m) þ (1 þ 2m)] J13 J31 J13 ¼ Q13 C11 (1 þ m) Q11 C13 g14 (1 þ m)[C11 S13 C13 S11 ]DV J31 ¼ Q31 C11 (1 þ m) Q11 C31 g14 (1 þ m)[C11 S31 C31 S11 ]DV S11 ¼ gP1 m2 þ gP2 n2 b2 , S13 ¼ g P1 m2 þ 9g P2 n2 b2 , S31 ¼ 9gP1 m2 þ g P2 n2 b2

(G:2)

in the above equations, for the case of uniform temperature rise C11 ¼ gT1 m2 þ g T2 n2 b2 , C13 ¼ g T1 m2 þ 9g T2 n2 b2 , C31 ¼ 9g T1 m2 þ g T2 n2 b2 C7 ¼ g T1 ,

C8 ¼ g T2 , C9 ¼ C10 ¼ 1:0

(G:3)

and for the case of in-plane parabolic temperature variation C11

T1 4 m2 g T2 g 5 g T1 4 4 þ 2 ¼ gT1 m þ gT2 n b þ þ T2 9 3p2 n2 p4 n4 g224 b 4 4 þ 4 4 þ n2 b2 g224 gT1 g5 gT2 2 2 3p m p m 2

2 2

229

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A007 Final Proof page 230 4.12.2008 11:11am Compositor Name: VAmoudavally

230

Appendix G

T1 4 m2 g T2 g 5 g T1 4 4 C13 ¼ g T1 m þ 9gT2 n b þ þ þ 2 T2 9 27p2 n2 81p4 n4 g224 b 4 4 þ 9n2 b2 g224 g T1 g5 g T2 þ 4 4 2 2 3p m p m 2 T 4 9m g T2 g 5 g T1 4 4 1 2 2 2 C31 ¼ 9gT1 m þ gT2 n b þ 2 þ þ T2 9 3p2 n2 p4 n4 g224 b 4 4 þ þ n2 b2 g224 g T1 g5 g T2 27p2 m2 81p4 m4 T0 4 4 gT2 g5 gT1 þ þ , C7 ¼ gT1 T1 9 5 b2 g224 T0 4 4 þ C8 ¼ gT2 þ b2 g224 g T1 g5 gT2 T1 9 5 C5 3 4 p2 3 C6 3 4 p2 3 (G:4) p 2 4 , C10 ¼ 1 p 2 4 C9 ¼ 1 24p 5 n 24p 5 m n m 2

2 2

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A008 Final Proof page 231 4.12.2008 11:12am Compositor Name: VAmoudavally

Appendix H In Equation 3.76 1 T (S0 , S2 , S4 ), d(0) x ¼ g 24 lp dx , 4b2 g 14 C11 1 ¼ C11 (1 þ 2m), 32b2 4 1 m n4 b4 2 (1 þ m)2 (1 þ 2m)2 ¼ g14 g 24 C11 þ J31 J13 g 224 256b2

(2) (4) (l(0) p , lp , l p ) ¼

d(2) x d(4) x

(H:1)

and in Equation 3.77 (2) (4) (l(0) T , l T , lT ) ¼

1 (S0 , S2 , S4 ) g 14 C11

(H:2)

in which [with g08, g138, g318, etc. are deﬁned as in Appendix C] S0 ¼

Q11 1 g14 1 g214 , S2 ¼ Q2 (1 þ 2m), S4 ¼ C11 (C24 C44 ) (1 þ m) 16 g24 256 g224

Q11 ¼ g08 , C24 C44

Q13 ¼ g138 , Q31 ¼ g318 4 m g224 n4 b4 2 2 ¼ 2(1 þ m) (1 þ 2m) Q2 þ J 13 J31 8 m g4 n8 b8 ¼ (1 þ m)(1 þ 2m)[2(1 þ m)2 þ (1 þ 2m)] þ 24 J 13 J31

J13 ¼ Q13 C11 (1 þ m) Q11 C13 , J31 ¼ Q31 C11 (1 þ m) Q11 C31

(H:3)

in the above equations, for the case of compressive postbuckling Q2 ¼ m4 þ g 224 n4 b4 , C11 ¼ C13 ¼ m2 ,

dTx ¼

1 2 g 24 gT1 g5 gT2 DT , 4b g24 2

C31 ¼ 9m2

(H:4)

and for the case of thermal postbuckling due to heat conduction

231

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A008 Final Proof page 232 4.12.2008 11:12am Compositor Name: VAmoudavally

232

Appendix H Q2 ¼

(3g224 g25 )(m4 þ g 224 n4 b4 ) þ 4g5 g224 m2 n2 b2 g224 g25

C11 ¼ (gT1 m2 þ gT2 n2 b2 ), C31 ¼ (9gT1 m2 þ gT2 n2 b2 )

C13 ¼ (gT1 m2 þ 9gT2 n2 b2 ), (H:5)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A009 Final Proof page 233 4.12.2008 11:12am Compositor Name: VAmoudavally

Appendix I In Equation 4.19 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u u u u 2 3 qkl (qkl ) (pkl ) qkl (qkl )2 (pkl )3 t t 3 3 þ þ þ wkl ¼ þ 2 4 27 2 4 27 (k,l) 2 (k,l) (k,l) fkl ¼ c31 wkl þ c32 wkl þ c33 (k,l) (k,l) (k,l) (cx )kl ¼ c11 wkl þ c12 fkl þ c13 ,

(k,l) (k,l) (k,l) (cy )kl ¼ c21 wkl þ c22 fkl þ c23

(I:1)

where (k,l) (k,l) (k,l) , c12 , c13 ¼ c11

1 (k,l) (k,l) b43 b32

(k,l) (k,l) b42 b33

(k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) b41 b33 b31 b43 , b44 b33 b34 b43 , y3 b43 y4 b33 (k,l) (k,l) (k,l) c21 , c22 , c23 ¼

1 (k,l) (k,l) b42 b33

(k,l) (k,l) b43 b32 (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) b41 b32 b31 b42 , b44 b32 b34 b42 , y3 b42 y4 b32

(k,l) (k,l) (k,l) c31 , c32 , c33 ¼

1 (k,l) b24

þ

(k,l) (k,l) b22 c12

(k,l) (k,l) þ b23 c22

16g24 klb2 (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) , b21 þ b22 c11 þ b23 c21 , b22 c13 þ b23 c23 3p2 . (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) c12 c31 þ b13 c22 c31 þ b14 c31 s(k,l) c32 d1(k,l) ¼ b12 s(k,l) c31 (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) d2(k,l) ¼ b11 þ b12 c11 þ b12 c12 c32 þ b13 c21 þ b13 c22 c32 . (k,l) (k,l) (k,l) (k,l) þb14 c32 s(k,l) c33 s(k,l) c31 . (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) (k,l) d3(k,l) ¼ b12 c13 þ b13 c23 þ b14 c33 y1 s(k,l) c31 2 2 (k,l) 2 1 (k,l) (k,l) 3, qkl ¼ d3(k,l) þ d1 d2 d pkl ¼ d2(k,l) þ d1(k,l) 27 1 3

s(k,l) ¼

32g14 klb2 3p2

233

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234

Appendix I

(i,j)

b11 ¼ g 110 (im)4 þ 2g112 (im)2 ( jn)2 b2 þ g 114 ( jn)4 b4 g14 b2 (py m2 þ px n2 ) (i,j) b12 ¼ g 120 (im)3 þ g122 im( jn)2 b2 (i,j) (i,j) b13 ¼ g 131 (im)2 jnb þ g133 ( jn)3 b3 b14 ¼ g 14 g 140 (im)4 þ g142 (im)2 ( jn)2 b2 þ g 144 ( jn)4 b4 (i,j) b21 ¼ g24 g240 (im)4 þ g 242 (im)2 ( jn)2 b2 þ g244 ( jn)4 b4 (i,j) b22 ¼ g 24 g 220 (im)3 þ g222 (im)( jn)2 b2 (i,j) b23 ¼ g 24 g 231 (im)2 ( jn)b þ g233 ( jn)3 b3 (i,j)

b24 ¼ m4 þ 2g 212 (im)2 ( jn)2 b2 þ g214 ( jn)4 b4 (i,j)

b31 ¼ g 31 (im) g310 (im)3 g 312 (im)( jn)2 b2 (i,j)

b32 ¼ g 31 þ r320 (im)2 þ g 322 ( jn)2 b2 (i,j)

b33 ¼ g 331 (im)( jn)b (i,j) b34 ¼ g24 g220 (im)3 þ g 222 (im)( jn)2 b2 (i,j)

b41 ¼ g 41 ( jn)b g411 (im)2 ( jn)b g 413 ( jn)3 b3 (i,j)

b42 ¼ g 331 (im)( jn)b (i,j)

b43 ¼ g 41 þ r430 (im)2 þ g 432 ( jn)2 b2 (i,j) b44 ¼ g14 g231 (im)2 ( jn)b þ g 233 ( jn)3 b3 . .

(k,l) 2 (0) y1(k,l) ¼ M(0) ¼ S(0) l, y4(k,l) ¼ bS(0) k x k l þ b My l k, y3 x y

(I:2)

and in Equation 4.20 "

M(0) x

S(0) x

M(0) y

S(0) y

# ¼

"

gT3

(g T3 gT6 )

* D22 * A11 * A22 * 1=4 gT4 p2 ½D11

(g T4 gT7 )

16hT1

# (I:3)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A010 Final Proof page 235 9.12.2008 6:40pm Compositor Name: VBalamugundan

Appendix J In Equations 4.31 through 4.35 (with i, j ¼ 1, 3) ( 2 1 g 24 g T1 g 5 g T2 þ g 5 (g T2 g 5 g T1 ) T1 g 25 g 224 b2 4 X X 1 2 g 5 g 224 n2 b2 fkl g 24 (g 511 þ g 5 g 220 )mckl 2 p k¼1,3,... l¼1,3,... kl

B(0) 00 ¼

g 24 (g 233 þ g 5 g 522 )nbckl þ g 24 g 611 m2 þ g 244 n2 b2 wkl þ g 5 g 24 g 240 m2 þ g 622 n2 b2 wkl b(0) 00

)

( 2 1 g 5 g 24 g T1 g 5 g T2 þ g 224 (g T2 g 5 g T1 ) T1 g 25 g 224 4 X X 1 2 g 5 g 224 m2 fkl g 24 g 5 g 511 þ g 224 g 220 mckl 2 p k¼1,3,... l¼1,3,... kl

¼

) 2 2 2 2 3 2 2 2 g 24 g 5 g 233 þ g 24 g 522 nbckl þ g 5 g 24 g 611 m þ g 244 n b wkl þ g 24 g 240 m þ g 622 n b wkl (i,j)

(i,j) (i,j) (i,j) (i,j) (i,j) (i,j) k k31 k32 k21 (i,j) , g21 ¼ 22 (i,j) (i,j) (i,j) (i,j) (i,j) (i,j) (i,j) (i,j) k22 k33 k32 k23 k23 k32 k33 k22 (i,j) (i,j) (i,j) (i,j) (i,j) ¼ a1 þ b1 g11 þ c1 g21 g 24 m2 n2 b2 , ¼ 2 16g 214 n4 b4 þ 64g 14 g 24 g 2223 n6 b6 = g 41 þ 4g 432 n2 b2 g 24 m2 n2 b2 ¼ 4 2 16m þ 64g 14 g 24 g 2220 m6 = g 31 þ 4g 320 m2 8g 220 g 14 m3 8g 233 g 14 n3 b3 ¼ g420 , g22 ¼ g402 2 g 31 þ 4g 320 m g 41 þ 4g 432 n2 b2 2 2 2 ¼ 8g12 g 120 m3 þ 16g420 g 14 g 140 m4 þ g(1,1) 31 g 14 b m n 2 2 2 ¼ 8g22 g 133 n3 b3 þ 16g420 g 14 g 144 n4 b4 þ g(1,1) 31 g 14 b m n 2 2 2 2 2 2 2 g (m þ g 5 n b ) (2) g 24 g 5 m þ g 24 n b ¼ 24 2 , b00 ¼ 8 g 25 g 224 8 g 5 g 224 b2

g11 ¼ (i,j)

g31

g402 g420 g12 g441 g442 B(2) 00

(i,j) (i,j)

k23 k31 k33 k21

2 2 (2) 2 2 g 14 b2 m2 B(2) 00 þ n b00 2m n g 14 b (g402 þ g420 ) g311 ¼ (1,1) (1,1) (1,1) (1,1) (1,1) 2 (0) (1,1) k11 þ k12 g11 þ k13 g21 g 14 b2 m2 B(0) 00 þ n b00 þ s g331 ¼

2g 14 m2 n2 b2 g420 (3,1) (3,1) (3,1) (3,1) (3,1) 2 (0) (3,1) k11 þ k12 g11 þ k13 g21 g 14 b2 9m2 B(0) 00 þ n b00 þ s

g313 ¼

2g 14 m2 n2 b2 g402 (1,3) (1,3) (1,3) (1,3) (1,3) 2 (0) (1,3) k11 þ k12 g11 þ k13 g21 g 14 b2 m2 B(0) 00 þ 9n b00 þ s

235

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A010 Final Proof page 236 9.12.2008 6:40pm Compositor Name: VBalamugundan

236

Appendix J

(1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) b2 k13 k12 k33 k32 k13 þ b3 k13 k12 k23 k22 k13 g1 ¼ (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) k11 k23 k21 k13 k33 k32 k13 k11 k33 k31 k13 k23 k22 k13 k12 k12 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) b2 k13 k11 k33 k31 k13 þ b3 k13 k11 k23 k21 k13 g2 ¼ (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) k12 k23 k22 k13 k33 k31 k13 k12 k33 k32 k13 k23 k21 k13 k11 k11 g3 ¼

(1,1) (1,1) k11 g1 þ k12 g2 (1,1) k13

(1,1) (1,1) , g4 ¼ a(1,1) 1 g1 þ b1 g2 þ c1 g3

(1,1) (1,1) (1,1) (1,1) (1,1) 2 (0) (1,1) g41 ¼ k11 þ k12 g11 þ k13 g21 g 14 b2 m2 B(0) 00 þ n b00 þ s h i 2 (2) 2 2 g42 ¼ g 14 b2 m2 B(2) 00 þ n b00 2m n (g402 þ g420 ) (1,1) g43 ¼ g 170 g 171 m2 g 171 n2 b2 g 80 mg(1,1) 11 g 80 nbg21

2 2 1 g441 (1 cos mp) þ cos 3mp cos mp g44 ¼ 2 p mn 3 3 2 1 (2) 2 2 þ g442 (1 cos np) þ cos 3np cos np g 14 b2 B(2) 00 m þ b00 n wmn 3 3

(J:1)

where

2 X X 1 1 1 1 1 1 2 2 2 2 2(k n þ l m ) p2 k¼1,3,... l¼1,3,... k 2k þ 4m 2k 4im l 2l þ 4n 2l 4n 1 1 1 1 (k,l) wkl þ fkl þ þ g13 klmn 2m þ k 2m k 2n þ l 2n l

s(m,n) ¼

b2 ¼ g90 m þ g 10 g(1,1) 11 ,

b3 ¼ g 90 nb þ g 10 g(1,1) 21

( J:2)

In the above equations (i,j) k11 ¼ g110 (im)4 þ 2g112 (im)2 (jn)2 b2 þ g 114 ( jn)4 b4 þ g14 g140 (im)4 þ g142 (im)2 (jn)2 b2 (i,j) (0) 2 2 þg144 ( jn)4 b4 a1 a1 g14 b2 B(0) 00 m þ b00 n (i,j) (i,j) k12 ¼ g120 (im)3 þ g122 im( jn)2 b2 þ g 14 g140 (im)4 þ g142 (im)2 ( jn)2 b2 þ g144 ( jn)4 b4 b1 (i,j) (i,j) k13 ¼ g131 (im)2 jnb þ g133 ( jn)3 b3 þ g14 g140 (im)4 þ g 142 (im)2 ( jn)2 b2 þ g144 ( jn)4 b4 c1 (i,j) (i,j) k21 ¼ g31 im g310 (im)3 g312 im( jn)2 b2 g14 g220 (im)3 þ g222 (im)( jn)2 b2 a1 (i,j) (i,j) k22 ¼ g31 þ r320 (im)2 þ g322 ( jn)2 b2 g14 g220 (im)3 þ g 222 (im)( jn)2 b2 b1 (i,j) (i,j) k23 ¼ g331 (im)( jn)b g 14 g220 (im)3 þ g222 (im)( jn)2 b2 c1 (i,j) (i,j) k31 ¼ g41 jnb g 411 (im)2 jnb g 413 ( jn)3 b3 g14 g231 (im)2 ( jn)b þ g233 ( jn)3 b3 a1 (i,j) (i,j) k32 ¼ g331 (im)( jn)b g 14 g231 (im)2 (jn)b þ g 233 ( jn)3 b3 b1 (i,j) (i,j) k33 ¼ g41 þ r430 (im)2 þ g432 ( jn)2 b2 g14 g231 (im)2 ( jn)b þ g233 ( jn)3 b3 c1 ( J:3)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A010 Final Proof page 237 9.12.2008 6:40pm Compositor Name: VBalamugundan

Appendix J

237

in which, when wkl, cxkl, cykl and fkl are considered, a1 ¼ 1, otherwise a1 ¼ 0, and g24

(i,j)

a1 ¼

m4

(

(i,j)

b1

(i,j)

c1

2

þ 2g 212 (im) ( jn)2 b2 þ g214 ( jn)4 b4

g240 (im)4 þ g 242 (im)2 ( jn)2 b2 þ g244 ( jn)4 b4 þ

2b2 X X p2 k¼1,3... l¼1,3,...

1 1 1 1 1 1 2(k2 n2 þ l2 m2 ) k 2k þ 4m 2k 4m l 2l þ 4n 2l 4n ) 1 1 1 1 þ wkl mnkl 2m þ k 2m k 2n þ l 2n l g 24 g 220 (im)3 þ g222 (im)( jn)2 b2 ¼ m4 þ 2g 212 (im)2 ( jn)2 b2 þ g214 ( jn)4 b4 g24 g231 (im)2 ( jn)b þ g233 ( jn)3 b3 ¼ ( J:4) m4 þ 2g 212 (im)2 ( jn)2 b2 þ g214 ( jn)4 b4

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A010 Final Proof page 238 9.12.2008 6:40pm Compositor Name: VBalamugundan

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A011 Final Proof page 239 4.12.2008 11:14am Compositor Name: VAmoudavally

Appendix K In Equations 4.73 through 4.76 (with other symbols can be found in Appendix J)

g1 ¼ g2 ¼ g41 ¼ g42 ¼

(1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) b3 k12 k23 k22 k13 k33 k13 k32 b2 k12 gxxk3 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) b2 k11 k33 k13 k31 k23 k13 k21 b3 k11 gxxk3 (1,1) (1,1) (1,1) (1,1) þ k12 g11 þ k13 g21 2g 14 b2 m2 n2 (g402 þ g420 ) (1,1) k11

g14 b2 m2 gxxk1 þ n2 gxxk2 þ s(1,1)

2 2 1 g (1 cos np) þ cos 3mp cos mp 441 p2 mn 3 3 2 1 þ g442 (1 cos mp) þ cos 3np cos np 3 3

g44 ¼

(K:1)

where s

(i,j)

1 1 1 1 1 1 2(k n þ l m ) k 2k þ 4m 2k 4m l 2l þ 4n 2l 4n 1 1 1 1 þ þ «A(1) mnkl 11 2m þ k 2m k 2n þ l 2n l 1 1 1 1 1 1 2 2 2 2 þ 2(k n þ 9l m ) k 2k þ 4m 2k 4m 3l 6l þ 4n 6l 4n 1 1 1 1 þ þ «3A(3) 3mnkl 13 2m þ k 2m k 2n þ 3l 2n 3l 1 1 1 1 1 1 þ 2(9k2 n2 þ l2 m2 ) 3k 6k þ 4m 6k 4m l 2l þ 4n 2l 4n 1 1 1 1 (i,j) g31 þ þ «3A(3) 3mnkl 31 2m þ 3k 2m 3k 2n þ l 2n l

2 ¼ 2 p

2 2

2

2

239

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A011 Final Proof page 240 4.12.2008 11:14am Compositor Name: VAmoudavally

240

Appendix K

(0) 2 (2) 4 (4) 2 (2) 4 (4) gxxk1 ¼ B(0) 00 þ « B00 þ « B00 , gxxk2 ¼ b00 þ « b00 þ « b00 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) gxxk3 ¼ k11 k22 k33 þ k12 k23 k31 þ k21 k32 k13 k23 k32 k11 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) k22 k31 k13 k21 k12 k33 g24 gxxk4 ¼ m4 þ 2g212 (im)2 ( jn)2 b2 þ g 214 ( jn)4 b4 (i,j)

k11 ¼ g110 (im)4 þ 2g112 (im)2 ( jn)2 b2 þ g 114 ( jn)4 b4 (i,j)

þ g14 ½g140 (im)4 þ g 142 (im)2 ( jn)2 b2 þ g 144 ( jn)4 b4 a1 (i,j) a1 ¼ gxxk4 g 240 (im)4 þ g 242 (im)2 ( jn)2 b2 þ g 244 ( jn)4 b4 2b2 1 1 1 þ gxxk4 2 2(k2 n2 þ l2 m2 ) p k 2k þ 4m 2k 4m 1 1 1 1 1 1 1 mnkl þ þ «A(1) 11 l 2l þ 4n 2l 4n 2m þ k 2m k 2n þ l 2n l 1 1 1 1 1 1 þ 2(k2 n2 þ 9l2 m2 ) k 2k þ 4m 2k 4m 3l 6l þ 4n 6l 4n 1 1 1 1 3mnkl þ þ «3A(3) 13 2m þ k 2m k 2n þ 3l 2n 3l 1 1 1 1 1 1 þ 2(9k2 n2 þ l2 m2 ) 3k 6k þ 4m 6k 4m l 2l þ 4n 2l 4n 1 1 1 1 3 (3) (K:2) 3mnkl þ þ « A31 2m þ 3k 2m 3k 2n þ l 2n l

and A(1) 11 « may be solved from 2 4 (2) (1) (4) (1) li ¼ l(0) þ l A « þ l A « þ 11 11 i i i

(i ¼ p, T)

in which (with other symbols can be found in Appendices F and G) (2) (4) l(0) ¼ p , lp , lp

1 1 (0) (2) (4) (S , S , S ), l , l , l (S0 , S2 , S4 ) ¼ 0 2 4 T T T 2 g 4b g 14 C11 14 C11 Q11 1 g 14 1 g 214 , S2 ¼ Q2 (1 þ 2m), S4 ¼ C11 C44 S0 ¼ (1 þ m) 16 g 24 256 g 224 3 3 m4 n4 b4 3 (3) «3 A(3) C11 A(1) C11 A(1) (K:3) 13 ¼ 11 « , « A31 ¼ 11 « 16J13 16J31

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A012 Final Proof page 241

6.12.2008 3:49pm Compositor Name: DeShanthi

Appendix L In Equations 5.51 through 5.54 1 m4 (1 þ m) 1 m2 g11 2g5 (2) g g « g24 g14 þ l Q1 ¼ C3 14 24 16n2 b2 g09 g06 32n2 b2 g09 g 24 p Q2 ¼

1 2g ½ðgT2 g5 g T1 ÞDT þ 5 l(0) g24 p g 24

( 1 g24 m2 g05 þ (1 þ m)g07 ¼ «1 þ g 24 2 (1 þ m)g06 (1 þ m)2 g06 1 g05 (1 þ m)g07 m(2 þ m)g05 « g08 þ g14 g 24 þ g06 g 14 (1 þ m)m2 (1 þ m)2 m g05 g05 1þ « 2 g m4 (1 þ m)m2 (1 þ m) 14 g05 (1 þ m)g07 þ g05 g08 þ g14 g 24 (2 þ m) «2 g06 (1 þ m)2 1 m6 (2 þ m) 1 m4 2 g 14 g224 ¼ « þ g g 14 24 2g09 g06 8 2g09 g206 g05 g07 1 g11 þ (1 þ m) þ g12 (1 þ m) g06 g06 1þm 2 1 1 g07 2 2 m g11 g05 g 24 m g13 (1 þ 2m)« þ g14 g 24 g12 « 2g09 g06 1 þ m g06 4 " # m2 g05 2(1 þ m)2 (1 þ 2m) m g05 þ g14 g224 g14 þ 2 2g09 g06 1 þ m g06 2(1 þ m) m2 n4 b4 S2 « (2 þ m)« þ g24 g06 S1

l(0) p

l(2) p

1 2 3 m10 (1 þ m) S3 1 g g « 128 14 24 g209 g306 S13 2 1 2 g 25 (2) g5 b11 1=2 2 1=2 « l lp ¼ g 224 qb01 fb(2) þ « p 10 g24 p g24 2pg224 q 1 2 ¼ g24 gT1 g 5 g T2 DT 2g24

l(4) p ¼ d(0) x d(T) x

241

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A012 Final Proof page 242

6.12.2008 3:49pm Compositor Name: DeShanthi

242 d(2) x d(4) p

Appendix L 1 g205 3 2 2 ¼ m (1 þ 2m)« 2g05 « þ 2 « m 16 ( 2 ) 1 b11 2 2 m8 (1 þ m)2 3=2 2 S4 2 4 4 g g ¼ « þ m n b (1 þ m) «3 S1 128 32pq 14 24 n4 b4 g209 g206

(L:1)

in the above equations S1 ¼ g06 (1 þ m) 4m2 C2 g10 , S13 ¼ g136 C9 g06 (1 þ m) S2 ¼ g06 5 þ 11m þ 4m2 þ 8m4 (1 þ m)(2 þ m)g10 S3 ¼ g136 6 þ 6m þ m2 þ g06 6 m2 (1 þ m) S4 ¼ g06 (1 þ 2m) þ 8m4 (1 þ m)g10 g05 q (1) g17 , b(2) «, a(1) 01 ¼ 1, a10 ¼ 01 ¼ g 24 g19 , 2 m f 4 1 (1) 2 2 (1) 2 2 4 a10 f b þ a10 2qfc þ 2q q f þ f ¼ b

C3 ¼ 1 b11

b(2) 10 ¼ g 24

q g20 f (L:2)

and g00 ¼ g 31 þ g 320 m2 þ g 322 n2 b2 g41 þ g430 m2 þ g432 n2 b2 g2331 m2 n2 b2 g01 ¼ g 41 þ g 430 m2 þ g 432 n2 b2 g220 m2 þ g222 n2 b2 g331 n2 b2 g 231 m2 þ g 233 n2 b2 g02 ¼ g 31 þ g 320 m2 þ g 322 n2 b2 g231 m2 þ g233 n2 b2 g331 m2 g 220 m2 þ g 222 n2 b2 g03 ¼ g 31 þ g 320 m2 þ g 322 n2 b2 g41 g411 m2 g413 n2 b2 g 331 m2 g31 g310 m2 g312 n2 b2 g04 ¼ g 41 þ g 430 m2 þ g 432 n2 b2 g31 g310 m2 g312 n2 b2 g 331 n2 b2 g41 g411 m2 g413 n2 b2 g05 ¼ g 240 m4 þ 2g 242 m2 n2 b2 þ g 244 n4 b4 m2 g 220 m2 þ g 222 n2 b2 g04 þ n2 b2 g231 m2 þ g233 n2 b2 g03 þ g00

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A012 Final Proof page 243

6.12.2008 3:49pm Compositor Name: DeShanthi

Appendix L

243

g06 ¼ m4 þ 2g212 m2 n2 b2 þ g 214 n4 b4 m2 g 220 m2 þ g 222 n2 b2 g01 þ n2 b2 g231 m2 þ g233 n2 b2 g02 þ g 14 g24 g00 4 2 2 2 4 4 g07 ¼ g140 m þ 2g142 m n b þ g 144 n b m2 g 120 m2 þ g 122 n2 b2 g01 þ n2 b2 g 131 m2 þ g 133 n2 b2 g02 g00 4 2 2 2 g08 ¼ g110 m þ 2g112 m n b þ g 114 n4 b4 m2 g 120 m2 þ g 122 n2 b2 g04 þ n2 b2 g 131 m2 þ g 133 n2 b2 g03 þ g00 4m2 g10 ¼ 1 þ g14 g 24 g2220 g 31 þ g320 4m2 g244 g41 þ g 432 4n2 b2 þ g 233 g 41 g413 4n2 b2 g12 ¼ g214 g 41 þ g432 4n2 b2 þ g14 g 24 g2233 4n2 b2 g214 g41 g 413 4n2 b2 g 14 g24 g 233 g 244 4n2 b2 *¼ g12 g214 g41 þ g 432 4n2 b2 þ g 14 g24 g 2233 4n2 b2 * þ g14 g 24 g144 g12 g09 ¼ g114 þ g133 g12 g 41 þ g432 4n2 b2 g214 g41 þ g 432 4n2 b2 þ g 14 g24 g 2233 4n2 b2 g 144 g 41 þ g432 4n2 b2 g133 g233 4n2 b2 ¼ g214 g41 þ g 432 4n2 b2 þ g 14 g24 g 2233 4n2 b2 g05 ¼ g14 (1 þ 2m) þ 2 g06 2 ¼ g31 þ g 320 m þ g 322 9n2 b2 g 41 þ g430 m2 þ g432 9n2 b2 g2331 9m2 n2 b2 ¼ g41 þ g 430 m2 þ g 432 9n2 b2 g 220 m2 þ g 222 9n2 b2 2 g 331 9n2 b2 g 231 m2 þ g 233 9n2 b ¼ g31 þ g 320 m2 þ g 322 9n2 b2 g 231 m2 þ g 233 9n2 b2 g 331 m2 g220 m2 þ g 222 9n2 b2 ¼ m4 þ 18g212 m2 n2 b2 þ g 214 81n4 b4 m2 g 220 m2 þ g 222 9n2 b2 g131 þ 9n2 b2 g 231 m2 þ g 233 9n2 b2 g132 þ g 14 g24 g130

g13 ¼ g14 g11 g130 g131 g132 g136

g15 ¼ g220 ðg310 þ g120 Þ g320 ðg140 þ g240 Þ g16 ¼ g320 þ g14 g 24 g2220 ðg320 g110 g310 g120 Þ þ g 14 g24 ðg320 g 140 g 120 g 220 Þðg 320 g 240 g 310 g 220 Þ g17 ¼

ðg310 þ g14 g 24 g220 g240 Þb g 14 g24 g 220 ðg310 þ g14 g 24 g220 g240 Þb þ g 14 g24 g 220

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A012 Final Proof page 244

244

6.12.2008 3:49pm Compositor Name: DeShanthi

Appendix L

2 2q g17 c g 320 g g g320 g 240 g19 ¼ þ 310 220 2 2 b g 320 þ g14 g 24 g220 g320 þ g14 g24 g2220 2 2f g17 þ c g320 2 g320 g17 g20 ¼ þ b g 320 þ g14 g 24 g2220 b2 g 320 þ g14 g 24 g2220

2g 310 g320 ðg310 g 220 g17 g320 g240 Þ½ðg 310 þ g 14 g24 g 220 g 240 Þb þ g 14 g24 g 220 g 320 þ g14 g 24 g2220 ½ðg 310 þ g14 g24 g220 g 240 Þb þ g 14 g24 g 220 (L:3)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A013 Final Proof page 245 4.12.2008 11:15am Compositor Name: VAmoudavally

Appendix M In Equation 5.87 through 5.90 1 1 1 C2 þ 1 ag 5 l(2) q C3 g24 2 1 1 1 ½ðgT2 g5 g T1 ÞDT« 1 ag 5 l(0) Q4 ¼ q g 24 g24 2 g24 m4 g 24 m2 g05 þ (1 þ m)g07 l(0) þ « q ¼ C1 (1 þ m)g06 C1 (1 þ m)2 g06 1 g05 (1 þ m)g07 mg05 2 g08 þ g 14 g24 þ« g06 g 14 C1 (1 þ m) (1 þ m)2 mg05 mg05 « 1 « 1 (1 þ m)m2 (1 þ m)m2 1 m4 n2 b2 1 g 24 g06 g13 ¼ 4g 24 (1 þ m) þ (1 þ 2m) l(2) q 4 g06 4 n2 b2 C1 g 24 n2 b2 g06 1 am2 2(1 þ m)2 þ (1 þ 2m) 6 C1 (1 þ m)g06 2am g10 2 C1 (1 þ 2m)g06 þ 8m4 g10 (1 þ m) þ2 g06 1 1 2 2 g5 1 (5=2) (5=2) 1=2 þ qb lq d(0) ¼ g 1 fb ag ag « 5 x 01 10 g 24 2 24 p g24 2 5 " 2 # 1 b11 1 1 ag5 « l2q þ 2 p(3)3=4 g 224 q Q3 ¼

(3)3=4 2 g 24 gT1 g5 gT2 DT «1=2 4g24 1 g2 ¼ (3)3=4 m2 (1 þ 2m)«3=2 2g05 «1=2 þ 052 «1=2 m 32

d(T) x ¼ d(2) x

(M:1) in the above equations 1 g05 C1 ¼ n2 b2 þ am2 , C3 ¼ 1 2 « m 2 1 2 2 g06 (1 þ 2m) þ 8m4 (1 þ m)g10 1 C2 ¼ n b (1 þ 2m) þ C1 (1 þ m) C1 (1 þ m)2 C1 (1 þ m)g06 2am6 g10 8 4

245

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A013 Final Proof page 246 4.12.2008 11:15am Compositor Name: VAmoudavally

246

Appendix M (3=2)

a01

b11

q q (3=2) (5=2) (5=2) ¼ 1, a10 ¼ g17 , b01 ¼ g 24 g19 , b10 ¼ g 24 g20 f f

1 (3=2) 2 2 (3=2) a10 ¼ f b þ a10 2qfc þ 2q4 q2 f2 þ f4 b

(M:2)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 247

6.12.2008 6:43pm Compositor Name: DeShanthi

Appendix N In Equations 5.122 through 5.126 1 2g 5 (0) 1=4 [(gT2 g 5 g T1 )DT] þ lp « g 24 g 24 4 1 g 14 g24 m (1 þ m) 1 g14 g 24 m2 g11 Q6 ¼ « C3 16n2 b2 g09 g210 32n2 b2 g09 Q5 ¼

2 2 1 2 2 2g 5 (2) 1=4 2 2 g210 þ g220 « þ (n b þ k b ) (1 þ 2m)« þ l 8 g 24 p g2210 ( m g m2 g g (0) 2 24 2 220 ls ¼ «5=4 þ 24 2 2nb g210 g220 (1 þ m) (1 þ m) g2210 g2220

(N:1)

[(1 þ m)(g210 g32 þ g220 g31 ) þ (g220 g310 þ g210 g320 )]«1=4 1 g120 g24 g31 (g220 g310 þ g210 g320 ) þ g32 (g220 g320 þ g210 g310 ) þ 2 þ m (1 þ m) g14 (1 þ m) g2210 g2220 2 g m 2g210 g310 g320 þ g220 g310 þ g2320 «3=4 24 2 2 g2 g (1 þ m) 210 220 m g110 g320 þ g120 g310 g24 g31 2g210 g310 g320 þ g220 g2310 þ g2320 þ g14 (1 þ m) g2210 g2220 m4 (1 þ m)2 2 g g32 g210 g310 þ g2320 þ 2g220 g310 g320 þ 24 (1 þ m) g2 g2220 2 210 g 24 m g220 g310 g310 þ 3g2320 þ g210 g320 3g2310 þ g2320 «7=4 g2210 g2220 (1 þ m)2 g120 g2310 þ g2320 þ 2g110 g310 g320 m2 þ g14 m6 (1 þ m)3 2 2 g 24 g31 g220 g310 g310 þ 3g320 þ g210 g320 3g2310 þ g2320 þ (1 þ m) g2210 g2220 2 g g32 g220 g320 3g310 þ g2320 þ g210 g310 g2310 þ 3g2320 þ 24 (1 þ m) g2210 g2220 ) 2 g 24 m 4g210 g310 g320 g310 þ g2320 þ g220 g4310 þ 6g2310 g2320 þ g4320 11=4 « g2210 g2220 (1 þ m)2

247

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 248

6.12.2008 6:43pm Compositor Name: DeShanthi

248

Appendix N ( m g 14 g 224 m6 g210 g220 5=4 g 14 g 224 m4 « 2 2nb 2g09 g2 g2 2 8g09 g210 g2210 g2220 210 220

3g2210 þg2220 g220 (g310 g31 þ g2210 þ3g2220 g210 (g320 g32 ) «1=4 g14 g224 m4 g320 g2210 3g2220 1=4 « þ 2 2g09 g2210 g2220 (1þm)

l(2) s ¼

g g 2 m4 g220 14 24 g14 (1þ2m)g302 (1þm)2 «1=4 4g09 g2210 g2220 (1þm) g120 g2210 þg2220 þ2g110 g210 g220 mg 14 g 24 m2 g 14 8g09 g210 g2210 g2220 2 g g220 g310 þg2320 3g2210 þg2220 þ2g210 g310 g320 g2210 þ3g2220 þ 24 1þm g2210 g2220 g g210 g2210 þ3g2220 (g32 g310 þg31 g320 )þg220 (3g2210 þg2220 )(g31 g310 þg32 g320 ) 3=4 « þ 24 1þm g2210 g2220

g24 m2 g220 2g m2 n4 b4 g220 g13 (1þ2m)«3=4 þ 2 24 2 8 g210 g210 g220 (1þm) 2 g g2 [2(1þm)2 þ3(1þ2m)]þg210 g200 (1þm) 3=4 « 210 220 2 4 g210 g2220 (1þm)g210 g200

g14 g224 m2 g220 ½g210 g302 g11 4g310 g14 «3=4 8g09 g2210 g2220 g14 g224 m2 g320 g302 g2210 þg2220 3=4 g 14 g 224 m2 g14 (1þ2m) « þ 4g09 g210 g2210 g2220 8g09 g210 g2210 g2220 (1þm)2

g2210 þg2220 g320 þ2g210 g220 g310 «3=4 2 g 14 g 224 m2 g320 ð3g210 þg2220 Þg220 g320 þ(g2210 þ3g2220 )g210 g310 3=4 2 « g2210 g2220 2g09 g210 g210 g2220 (1þm) g14 g224 m2 3g2210 þg2220 g220 g11 (g31 g310 ) 3=4 « 2 2 16g09 g210 g2220 (1þm) ) g14 g224 m2 g2210 þ3g2220 g210 g11 (g32 g320 ) 3=4 « 2 16g09 g2210 g2220 (1þm)2

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 249

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Appendix N

l(4) s

249

( g220 g2210 þ 3g2220 m g214 g 324 m10 (1 þ m) g220 4 ¼ 2 2 2 2 2nb 64g209 g2210 g2220 g210 g210 g220

2 (1 þ m) 2 2 g24 g2210 g2220 þ 2g210 g220 g23 3g2210 þ g2220 þ 2 2 2 g210 g220 g23 g24 g210 2g2210 g24 g2210 þ 3g2220 (1 þ m) g220 g23 3g2210 g2220 g210 g24 g2210 þ g2220 þ g2210 g223 g224 g220 g210 g23 g220 g24 g220 R1 þ6 þ2 g210 g210 g223 g224 (1 þ m) g220 g24 3g2210 g2220 g210 g23 g2210 þ g2220 g2210 g223 g224 ) g220 g220 g23 g210 g24 g2210 þ g2220 R2 «5=4 þ (N:2) 2 g210 g223 g224 g2210

in which l(0) p ¼

kb (0) l þ l(0) xp , m s

l(2) p ¼

kb (2) l þ l(2) xp , m s

l(4) p ¼

kb (4) l þ l(4) xp , m s

(N:3)

and k can be determined through equation

2

4 (2) (2) (2) (4) l(0) l A « þ l A « ¼0 xp xp xp 11 11

(N:4)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 250

250

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Appendix N

where l(0) xp

1 ¼ 2

(

g m2 g g 2 24 2 210 «5=4 þ 24 [(1 þ m)(g210 g31 þ g220 g32 ) g210 g220 (1 þ m) (1 þ m)2 g2210 g2220

þ (g220 g320 þ g210 g310 )]«1=4 1 g110 g g31 (g220 g320 þ g210 g310 ) þ g32 (g220 g310 þ g210 g320 ) þ 2 þ 24 m (1 þ m) g14 (1 þ m) g2210 g2220 2 g m 2g220 g310 g320 þ g210 g310 þ g2320 «3=4 24 2 g2210 g2220 (1 þ m) m g110 g310 þ g120 g320 g24 g31 2g220 g310 g320 þ g210 g2310 þ g2320 þ g 14 (1 þ m) g2210 g2220 m4 (1 þ m)2 g g32 g220 (g2310 þ g2320 ) þ 2g210 g310 g320 þ 24 (1 þ m) g2 g2220 2210 g24 m g210 g310 g310 þ 3g2320 þ g220 g320 3g2310 þ g2320 «7=4 g2210 g2220 (1 þ m)2 g110 g2310 þ g2320 þ 2g120 g310 g320 m2 þ g 14 m6 (1 þ m)3 2 2 g 24 g31 g220 g320 3g310 þ g320 þ g210 g310 g2310 þ 3g2320 þ (1 þ m) g2210 g2220 2 g 24 g32 g220 g310 g310 þ 3g2320 þ g210 g320 3g2310 þ g2320 þ (1 þ m) g2210 g2220 ) 2 g24 m 4g220 g310 g320 g310 þ g2320 þ g210 g4310 þ 6g2310 g2320 þ g4320 11=4 « g2210 g2220 (1 þ m)2

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 251

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Appendix N

251

(

g14 g224 m6 g2210 þg2220 5=4 2 « 2 4g09 g2210 g2220

1 l(2) xp ¼

þ

2 g14 g224 m4 2 2 2 3g210 þg220 g220 (g320 g32 ) 2 8g09 g210 g210 g220

g2210 þ3g2220

1=4 g 14 g 224 m4 g220 g320 3g2210 þg2220 1=4 « þ g210 (g310 g31 ) « 2 2g09 g210 g2210 g2220 (1þm)

g14 g224 m4 g2210 þg2220 2 g14 (1þ2m)g302 (1þm)2 «1=4 8g09 g210 g210 g2220 (1þm)

g110 g2210 þg2220 þ2g120 g210 g220 mg 14 g24 m2 2 g 14 8g09 g210 g210 g2220

þ

g 24 g210 g2310 þg2320 g2210 þ3g2220 þ2g220 g310 g320 3g2210 þg2220 1þm g2210 g2220

g 24 g210 g2210 þ3g2220 (g31 þg310 þg32 g320 )þg220 (3g2210 þg2220 )(g32 g310 þg31 g320 ) 3=4 « þ 2 2 1þm g210 g220

2 4 4 2 n b g210 þg2220 g 24 m2 g2210 þg2220 3=4 g 24 m g (1þ2m)« þ 13 16 g2210 g210 g2210 g2220 (1þm)

g2210 g2220 [2(1þm)2 þ3(1þ2m)]þg210 g200 (1þm) 3=4 « 4 g2210 g2220 (1þm)g210 g200

g14 g 224 m2 g2210 þg2220 ½ g210 g302 g11 4g310 g14 «3=4 16g09 g210 g2210 g2220 þ

g 14 g 224 m2 g320 g302 g220 3=4 g 14 g 224 m2 g14 (1þ2m) 2 « 2 2g09 g210 g220 8g09 g210 g2210 g2220 (1þm)2

g2210 þg2220 g310 þ2g210 g220 g320 «3=4

2 g 14 g 224 m2 g320 ð3g210 þg2220 Þg220 g310 þ(g2210 þ3g2220 )g210 g320 3=4 2 « g2210 g2220 2g09 g210 g210 g2220 (1þm)

g 14 g 224 m2 3g2210 þg2220 g220 g11 (g32 g320 ) 3=4 « 2 2 16g09 g210 g2220 (1þm)

) g 14 g 224 m2 g2210 þ3g2220 g210 g11 (g31 g310 ) 3=4 « 2 16g09 g2210 g2220 (1þm)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 252

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Appendix N

l(4) xp

( g2220 3g2210 þ g2220 1 g 214 g324 m10 (1 þ m) 4 ¼ 2 1 2 64g209 g210 g2210 g2220 g2 g2 210

220

(1 þ m) þ 2 2 g210 g220 g223 g224 g210

g23 g2210 þ g2220 3g2210 þ g2220 2g24 g210 g220 2g2210 þ g2220 g2 þ g2220 g210 g23 g220 g24 þ (1 þ m) 210 g210 g223 g224 g210 g23 g220 g24 g2210 þ g2220 þ 2g210 R1 þ3 g210 g223 g224 g2 þ g2220 g220 g23 g210 g24 þ (1 þ m) 210 g210 g223 g224 ) g220 g23 g210 g24 2g220 R2 «5=4 þ 2g210 g223 g224

(N:5)

2 1 4 (9=4) (9=4) g 5 1=2 lp , 2g224 « qb01 fb10 g24 g 24 p 1 2 ¼ g 24 gT1 g5 gT2 DT «1=4 2g 24 1 b11 g2210 þ g2220 1=4 g2 þ g2 ¼ « þ m2 (1 þ 2m) 210 2 220 «3=4 2 8 2pq g g210 2 210 2 g310 g210 þ g220 2g320 g210 g220 7=4 2 « g2210 2 3 g þ g2220 g2310 þ g2320 4g210 g220 g310 g320 11=4 þ 2 210 « m g2210 2 g310 g2210 þ g2220 g2310 þ 3g2320 2g320 g210 g220 3g2310 þ g2320 15=4 4 « m g2210 ( 1 b11 g 214 g224 m8 (1 þ m)2 7=4 ¼ « 32 64pq n4 b4 g209 g2210

d(0) x ¼ d(T) x d(2) x

d(4) x

g 14 g24 g221 g222 (1 þ m)2 m4 (m2 g310 «)«3=4 g09 g210 g2210 2 2 g g2 þ 2m2 n4 b4 (1 þ m)2 210 2 220 g210 " #2 ) 2 g2210 g2220 (1 þ 2m) þ g210 g200 (1 þ m) 11=4 « 4 g2210 g2220 (1 þ m) g210 g200 þ

(N:6)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 253

6.12.2008 6:44pm Compositor Name: DeShanthi

Appendix N

g 266 ls g24 " 1 kb g2210 þ g2220 nb 2g220 m2 (1 þ 2m)«3=4 ¼ þ 4 m m g210 g2210 kb g310 g2210 þ g2220 2g320 g210 g220 m g2210 2 nb g320 g210 þ g2220 2g310 g210 g220 7=4 þ « m g2210 kb g2210 þ g2220 g2310 þ g2320 4g210 g220 g310 g320 þ2 m m2 g2210 2 nb g210 þ g2220 g310 g320 g210 g220 g2310 þ g2320 «11=4 þ2 m m2 g2210 kb g310 g2210 þ g2220 g2310 þ 3g2320 2g320 g210 g220 3g2310 þ g2320 m m4 g2210 2 # nb g320 g210 þ g2220 3g2310 þ g2320 2g310 g210 g220 g2310 þ 3g2320 «15=4 þ m m4 g2210 ( 1 kb m6 (1 þ m)2 g2210 g2220 3=4 ¼ « 32 m g09 g210 g2210 g14 g 24 g2210 g2220 nb 2 4 kb (1 þ m) m g310 þ g320 «7=4 g09 g210 m m g2210 2 2 kb g g2 þ 4m2 n4 b4 (1 þ m)2 210 2 220 m g210 " #2 ) 2 g2210 g2220 (1 þ 2m) þ g210 g200 (1 þ m) 11=4 « (N:7) 4 g2210 g2220 (1 þ m) g210 g200

g (0) ¼ g (2)

g (4)

253

in the above equations g210 ¼ m4 þ 2g212 m2 k2 b2 þ n2 b2 þ g 214 k4 b4 þ 6k2 b2 n2 b2 þ n4 b4 e4 D01 þ e1 D03 þ e3 D02 þ e2 D04 þ g 14 g24 D

00 g220 ¼ 4kbnb g222 m2 þ g214 k2 b2 þ n2 b2 e4 D02 þ e1 D04 þ e3 D01 þ e2 D03 g 14 g24 D00

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 254

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Appendix N

g310 ¼ g240 m4 þ 2g242 m2 k2 b2 þ n2 b2 þ g 244 k4 b4 þ 6k2 b2 n2 b2 þ n4 b4 e1 D06 þ e2 D08 þ e3 D07 þ e4 D05 D00

e1 D08 þ e2 D06 þ e3 D05 þ e4 D07 g320 ¼ 4kbnb g 242 m2 þ g244 k2 b2 þ n2 b2 D00

g31 ¼ g140 m4 þ 2g142 m2 k2 b2 þ n2 b2 þ g 144 k4 b4 þ 6k2 b2 n2 b2 þ n4 b4 h4 D01 þ h3 D02 þ h1 D03 þ h2 D04 D00

h4 D02 þ h3 D01 þ h2 D03 þ h1 D04 2 g32 ¼ 4kbnb g 142 m þ g144 k2 b2 þ n2 b2 D00 4 4

4 2 2 2 2 2 2 2 2 2 g110 ¼ g110 m þ 2g112 m k b þ n b þ g 114 k b þ 6k b n b þ n4 b4 h1 D06 þ h2 D08 þ h3 D07 þ h4 D05 D00

h1 D08 þ h2 D06 þ h3 D05 þ h4 D07 g120 ¼ 4kbnb g 112 m2 þ g114 k2 b2 þ n2 b2 D00 2

4 64m2 g220 m2 þ g222 k2 b2 2 2 2 4 4 g200 ¼ 16 m þ 2g212 m k b þ g 214 k b þ g 14 g24 g31 þ g 320 4 m2 þ k2 b2 g 244 g 41 þ g 432 4n2 b2 þ g 233 g 41 g413 4n2 b2 g302 ¼ g214 g41 þ g432 4n2 b2 þ g 14 g24 g 2233 4n2 b2 g 214 g 41 g 413 4n2 b2 g 14 g24 g 233 g244 4n2 b2 g303 ¼ g 214 g 41 þ g 432 4n2 b2 þ g14 g24 g2233 4n2 b2 g09 ¼ g 114 þ g133 g303 þ g14 g 24 g144 g302 g41 þ g432 4n2 b2 g 214 g 41 þ g 432 4n2 b2 þ g 14 g24 g 2233 4n2 b2 g144 g 41 þ g432 4n2 b2 g133 g233 4n2 b2 ¼ g 214 g 41 þ g 432 4n2 b2 þ g 14 g24 g 2233 4n2 b2

g13 ¼ g14

g2210 g2220 g210 g310 g220 g320 (1 þ 2m) þ 2 2 g210 g2210

4 ¼ m þ 2g212 m2 9n2 b2 þ k2 b2 þ g214 k4 b4 þ 54k2 b2 n2 b2 þ 81n4 b4 d1 D131 þ c1 D132 þ a1 D133 þ b1 D134 þ g 14 g24 D

130 ¼ 12kbnb g212 m2 þ g214 k2 b2 þ 9n2 b2 d1 D132 þ c1 D131 þ a1 D134 þ b1 D133 þ g 14 g24 D130

g11 ¼ g14 g23

g24

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 255

6.12.2008 6:44pm Compositor Name: DeShanthi

Appendix N

255 h1 ¼ g210 g223 g224 g23 g2210 g2220 (1 þ m) h2 ¼ 3g220 g223 g224 g24 g2210 g2220 (1 þ m) h3 ¼ g223 g224 þ (g210 g23 g220 g24 )(1 þ m) h4 ¼ (g220 g23 g210 g24 )(1 þ m) h 3 h 1 þ h4 h 2 h h þ h4 h 1 , R2 ¼ 3 22 , 2 2 h 1 h2 h1 h22 g210 g310 g220 g320 kp « cos C3 ¼ 1 m2 g210 2n e1 r12 r13 r14 r11 e1 r12 r13 r14 r e r22 r23 r24 e r r r , D01 ¼ 2 22 23 24 , D02 ¼ 21 2 r e r32 r33 r34 e3 r32 r33 r34 31 3 r41 e4 r42 r43 r44 e4 r42 r43 r44 r11 r12 r13 e1 f1 r12 r12 e1 r14 r21 r22 r23 e2 f r r22 e2 r24 , D04 ¼ , D05 ¼ 2 22 f r r32 e3 r34 r31 r32 r33 e3 3 32 f4 r42 r42 e4 r44 r41 r42 r43 e4 f2 r12 r13 r14 r11 r12 r12 f1 r14 f1 r22 r23 r24 r21 r22 r22 f2 r24 , D07 ¼ , D08 ¼ r32 f3 r34 f4 r32 r33 r34 r31 r32 r41 r42 r42 f4 r44 f3 r42 r43 r44 a1 s12 s13 s14 b1 s12 s12 s13 s14 b1 s22 s23 s24 a s s22 s23 s24 , D131 ¼ , D132 ¼ 1 22 d s s32 s33 s34 c1 s32 s33 s34 1 32 c1 s42 s42 s43 s44 d1 s42 s43 s44 s11 s12 b1 s14 s12 a1 s14 s21 s22 a1 s24 s22 b1 s24 , D134 ¼ s s d s s32 c1 s34 31 32 1 34 s41 s42 c1 s44 s42 d1 s44 R1 ¼

r11 r21 D00 ¼ r31 r41 r11 r21 D03 ¼ r31 r41 r11 r21 D06 ¼ r31 r41 s11 s21 D130 ¼ s31 s41 s11 s21 D133 ¼ s31 s41

r13 r14 r23 r24 r33 r34 r43 r44 r13 r14 r23 r24 r33 r34 r43 r44 f2 r14 f1 r24 f4 r34 f3 r44 s13 s14 s23 s24 s33 s34 s43 s44

r11 ¼ r22 ¼ r34 ¼ r43 ¼ mnbg 331 , r12 ¼ r21 ¼ r33 ¼ r44 ¼ mkbg 331 r13 ¼ r24 ¼ g41 þ g430 m2 þ g432 k2 b2 þ n2 b2 r14 ¼ r23 ¼ 2kbnbg 432 , r31 ¼ r42 ¼ 2kbnbg 322 r32 ¼ r41 ¼ g31 þ g320 m2 þ g322 k2 b2 þ n2 b2

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A014 Final Proof page 256

256

6.12.2008 6:44pm Compositor Name: DeShanthi

Appendix N

e1 ¼ nb g 231 m2 þ g233 3k2 b2 þ n2 b2

e2 ¼ kb g 231 m2 þ g233 k2 b2 þ 3n2 b2

e3 ¼ 2mkbnbg 222 , e4 ¼ m g220 m2 þ g 222 k2 b2 þ n2 b2 ,

f1 ¼ nb g 41 g411 m2 g 413 3k2 b2 þ n2 b2

f2 ¼ kb g41 g 411 m2 g413 k2 b2 þ 3n2 b2

f3 ¼ 2mkbnbg 312 , f4 ¼ m g 31 g310 m2 g 312 k2 b2 þ n2 b2

h1 ¼ nb g 131 m2 þ g133 3k2 b2 þ n2 b2

h2 ¼ kb g 131 m2 þ g133 k2 b2 þ 3n2 b2 h3 ¼ 2mkbnbg 122

h4 ¼ m g120 m2 þ g 122 k2 b2 þ n2 b2 s11 ¼ s22 ¼ s34 ¼ s43 ¼ 3mnbg 331 , s12 ¼ s21 ¼ s33 ¼ s44 ¼ 3mkbg 331 s13 ¼ s24 ¼ g 41 þ g430 m2 þ g 432 k2 b2 þ 9n2 b2 , s14 ¼ s23 ¼ 6kbnbg 431 s31 ¼ s42 ¼ 6kbnbg322 , s32 ¼ s41 ¼ g31 þ g 320 m2 þ g322 k2 b2 þ 9n2 b2

a1 ¼ 3nb g 231 m2 þ g233 3k2 b2 þ 9n2 b2 ,

b1 ¼ kb g231 m2 þ g 233 k2 b2 þ 27n2 b2

c1 ¼ 6mkbnbg 222 , d1 ¼ m g 220 m2 þ g222 k2 b2 þ 9n2 b2 ¼ 1, a10

(9=4)

¼ b(3) 01 ¼ g 24 g19 , b10

b01

(5=4)

q g17 f

(5=4)

a01

b11 ¼

¼

(9=4)

¼ b(3) 10 ¼ g 24

q g20 f

1 (5=4) 2 2 (5=4) f b þ a10 2qfc þ 2q4 q2 f2 þ f4 a10 b

(N:8)

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A015 Final Proof page 257 4.12.2008 11:16am Compositor Name: VAmoudavally

Appendix O In Equations 5.137 and 5.138 1 m4 (1 þ m) 1 m2 g11 g14 g 24 « g g Q7 ¼ 24 14 2 C3 16n2 b g09 g06 32n2 b2 g09 1 g5 g205 3 g 224 g25 gT2 (2) 2 2 þ m (1 þ 2m)« 2g05 « þ 2 « þ l m g24 gT T 8 g8 g224 g 25 g T2 (0) l g 24 gT T

Q8 ¼

(O:1)

and (0) l(0) T ¼ 2lp

1 g 24 g2 m2 (1 þ 2m)« 2g05 «2 þ 052 «3 m 8 g8 ( 2 8 1 g 24 b11 2 2 m (1 þ m) 3=2 g g ¼ 2l(4) « p þ 64 g8 32pq 14 24 n4 b4 g209 g206 2 ) 4 g (1 þ 2m) þ 8m (1 þ m)g 06 10 2 þ m2 n4 b4 (1 þ m) «3 g06 (1 þ m) 4m4 g10

(2) l(2) T ¼ 2lp

l(4) T

(O:2)

(j)

where lp (j ¼ 0, 2, 4) are deﬁned by Equation L.1. In the above equations (with other symbols are deﬁned by Equations L.2 and L.3) 2 g 25 (2) 1=2 qb01 fb(2) 10 « p g 24 2 g 5 (2) 1=2 gT ¼ (g 224 gT1 g5 gT2 ) þ qb01 wb(2) 10 (g T2 g 5 g T1 )« p g24

C11 ¼

g8 , gT

g8 ¼ g224

(O:3)

257

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_A015 Final Proof page 258 4.12.2008 11:16am Compositor Name: VAmoudavally

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C006 Final Proof page 259 8.12.2008 2:47pm Compositor Name: BMani

Index A Asymptotic solutions boundary layer conditions, order 3=2, 174 boundary layer conditions, order 5=4, 188 Average end-shortening relationships, 53, 155

B Batdorf shell parameter, 153 Bending moment and central deﬂection, 30 Boundary conditions, 27 associated with in-plane behavior, 23–24 Boundary layer asymptotic solutions, 202 effect of, 185 solutions, 174, 186, 187, 201 Boundary layer-type equations, 184 Buckling loads, 58, 176 for sandwich plates, 90 thermal environmental condition, 193 under T-ID case, 193 uniaxial compressed, Si3N4=SUS304 plates, 63 uniaxial compressed, (FGM=P)S plates Si3N4=SUS304, 60 ZrO2=Ti-6Al-4V, 62 uniaxial compressed, (P=FGM)S plates Si3N4=SUS304, 59 ZrO2=Ti-6Al-4V, 61, 63 Buckling loads comparison, 89 Buckling shear stress, 192 Buckling temperature FGM hybrid plates Si3N4=SUS304 substrate, 74–75 ZrO2=Ti-6Al-4V substrate, 75 (FGM=SUS304=FGM) sandwich plates, 91 (FGM=Ti-6Al-4V=FGM) sandwich plates, 91

(FGM=P)S plates Si3N4=SUS304 substrate, 77 ZrO2=Ti-6Al-4V substrate, 79 (P=FGM)S plates Si3N4=SUS304 substrate, 76 ZrO2=Ti-6Al-4V substrate, 78

C Carbon nanotubes, simulation tests, 180 Ceramic materials, applications, 1 Chromium-molybdenum steel tube, 191 buckling shear stress thick cylindrical shells, 193 thin cylindrical shells, 192 Circular cylindrical shells Donnell theory, 149 geometry and coordinate system of, 149 Clamped boundary conditions, asymptotic solutions, 160 Classical laminated plate theory, 124 Classical plate theory, 100 CLPT. See Classical laminated plate theory Compressive and thermal buckling problem, 85, 86, 88 Compressive postbuckling equilibrium path, 89 CPT. See Classical plate theory

D Donnell theory assumptions of, 150 circular cylindrical shells, 149

F FEM. See Finite element method FGM cylindrical shells at ambient temperature, 203 259

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C006 Final Proof page 260 8.12.2008 2:47pm Compositor Name: BMani

260 axial compression, postbuckling behavior of asymptotic solutions, 160 boundary conditions, 154, 156, 161 boundary layer solutions, 157, 158, 159 buckling loads comparison, 164 closed condition, 156 end-shortening relationship, 155 postbuckling load–shortening curves, 165, 166 thermal conductivity, 153 thermal environments, 157 transverse dynamic load, 154 boundary layer theory, 149 boundary layer-type equations, 152–153 Donnell theory, 149–150 Kármán–Donnell-type nonlinear equations, 150–151 buckling loads comparison, 162, 163 buckling temperatures, 203 comparison, 203, 204 volume fraction index, 202 CC perfect, buckling temperatures, 204 dynamic stability analysis of, 183 external pressure, postbuckling behavior average end-shortening relationship, 170 boundary conditions, 170 boundary layer, effect of, 171 end-shortening relationship, 171 nonlinear differential equations, 169 perturbation equations, sets of, 172 nonlinear differential equations, 183 nonlinear postbuckling load– shortening, 162, 241–244, 247–256 Poisson ratio, 153 postbuckling behavior of, 183, 191, 201 postbuckling load–deﬂection curves, 204, 206 postbuckling solutions of, 147 thermal buckling of, 148 thermal effect, 185 thermal loading, 201

Index thermal postbuckling behavior comparison, 207 material properties, effect of, 205 volume fraction index, effect of, 206 thermal postbuckling behavior of average end-shortening relationship, 200 axial compression, 200 buckling problem for, 199 thermal postbuckling equilibrium path, 207 thermal postbuckling load–deﬂection curves, 202 numerical results, 257 transverse displacement, 151 volume fraction index, 203 FGM dental implant, 3 FGM face sheets asymptotic solutions, 136 heat transfer, 132 thermally postbuckled sandwich plates thickness of, 132 vibration of, 132 FGM hybrid cylindrical shells, 148 FGM hybrid laminated plates, 118 FGM hybrid plate natural frequency parameter, 125–126 volume fraction index, effect of, 127 FGM layer, 118 FGM plates bending–stretching coupling effect, 104 boundary conditions, 102, 103, 123 buckling analysis of axisymmetric, 45–46 bifurcation, 47 buckling loads, 46 thermal and mechanical, 45 dynamic transverse loads, 99 forced vibration, 108 free vibration equation, 108 geometrically midplane symmetric boundary conditions, 54–55 geometric imperfection, 55 nonlinear equations, 54 perturbation equations, 56–57 postbuckling equilibrium path, 57

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C006 Final Proof page 261 8.12.2008 2:47pm Compositor Name: BMani

Index stretching=bending coupling, 53 unit end-shortening relationships, 55 natural frequency, comparison of, 110 nonlinear bending of deﬂection, 21 under mechanical loads (see Nonlinear bending, FGM plates under mechanical loads) thermal loads, 22 transverse uniform=sinusoidal load, 22 nonlinear frequency of, 108, 137 nonlinear vibration of, 100 piezoelectric actuators, 118 perturbation technique, 105, 106 with piezoelectric actuator, 124 postbuckling analysis of hybrid laminated plate (see Hybrid laminated plate) thermal, 46, 47 thermomechanical loads, 46 rectangular, 22 shear deformable (See Shear deformable FGM plates) thermal bending, heat conduction bending moments, 38 in double Fourier sine series, 37 effect of temperature variation, 36–37 nonlinear governing equations, 37 thermal load–deﬂection, load– bending moment curves (see Thermal load-deﬂection and thermal load-bending moment curves) thermal environments, 102 thermal expansion, 101, 109 Young’s modulus, 109 FGM square plate aluminum oxide, 109 central deﬂection vs. time curves, 111 dynamic response of, 111 free vibration of, 124 natural frequency parameter comparisons, 112 thermal environments, 112–114

261 nonlinear to linear frequency ratios, 114 temperature ﬁeld, effect of, 115 volume fraction index effect, 115, 116, 117 FGM=SUS304=FGM sandwich plates buckling loads comparison, 90 buckling temperatures comparison, 91 postbuckling behavior of effect of material properties on, 92 effect of volume fraction index on, 94–95 thermal effects on, 93 thermal postbuckling behavior of effects of material properties on, 95 effects of volume fraction index on, 96 FGM thin-walled tapered pretwisted turbine blade, 2 FGM=Ti-6Al-4V=FGM sandwich plates buckling loads comparison, 90 buckling temperatures comparison, 91 Finite element method, 99 First-order shear deformation plate theory (FSDPT), 9–10 Functionally graded materials (FGMs) applications of, 2 material properties, approaches to model mechanical response, 5 position-dependent, 6–7 simple rule of mixture, 5 temperature-dependent, 5–6 volume fraction variation, 3–4 microstructural phases, 1 nonlinear static and dynamic response of, 21 rectangular plate displacement components, 9 strains associated with displacement ﬁeld, 10–11 transverse shear strains, 10 research studies in, 3 skeletal microstructure of, 8 as structural components, 2

Shen/Functionally Graded Materials: Nonlinear Analysis of Plates and Shells 92561_C006 Final Proof page 262 8.12.2008 2:47pm Compositor Name: BMani

262

Index (FGM=P)S, temperature effects, 69 temperature ﬁeld, 49 thermal postbuckling behavior asymptotic solutions, 71–72 boundary conditions, 70–71 edges of plate, 67 estimation procedure, 72–73 material properties effect, 78–79 temperature ﬁeld, 66 thermal load ratio effect, 80–81, 82 thermal postbuckling equilibrium path, 72 volume fraction index effect, 80, 81 transformed elastic constants, 50–51

as thermal barrier materials, 1 with volume fractions of constituent phases graded in one direction, 2

G Galerkin procedure, 104 General von Kármán-type equations hybrid laminated plate, 49 shear deformable FGM plates bending moments, 15 equations of motion, 15–17 higher order moments, 15 thermal effect, 14 transverse shear forces, 15

I H Hamilton principle, elastic body, 12 Higher order shear deformation plate theory (HSDPT), 9–10 Homogeneous isotropic cylindrical shells, 157 Hybrid FGM plates applied voltage, effect of, 129, 130 temperature ﬁeld, effect of, 128, 129 types of, 124 volume fraction index, effect of, 127, 128 Hybrid laminated plates components of, 47–48 conﬁgurations of, 48 edges, simply supported associated boundary conditions, 51–52 immovability condition, 52 stretching=bending coupling, 53 electric ﬁeld forces and moments caused by, 49–50 transverse direction component, 49 load applied to, 49 (P=FGM)S, postbuckling behaviour temperature rise effects, 68 thermopiezoelectric effects, 62, 64–65 volume fraction index effect, 63, 66, 67

Isotropic cylindrical shells, 191 buckling loads comparison, 177, 191 buckling shear stress comparison, 191 hydrostatic pressure, 176, 177 postbuckling load–shortening curves, 162, 163 Isotropic rectangular plate, 137 Isotropic square plate, linear frequency ratios, 110

K Kirchhoff–Love hypotheses, 150

L Lightweight composite materials, applications, 1 Load–bending moment curves zirconia=aluminum square plate under in-plane boundary conditions, 33–34 thermal environmental conditions, 33 transverse loading conditions, 34–35 uniform pressure conditions, 31, 33 volume fraction index, 32 Load-central deﬂection curves relationship, expression for, 30

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Index zirconia=aluminum square plate comparisons of, 31 in-plane boundary conditions effect, 34 temperature-dependency effect, 40 temperature rise effect, 33 uniform=sinusoidal load, thermal environments, 35 volume fraction index effect, 32

M Micromechanics models, 8 Mori–Tanaka model, 7, 9

N Nondimensional compressive stress, 190 Nonlinear bending, FGM plates under mechanical loads transverse static load, 23 transverse uniform=sinusoidal loads, 22 Nonlinear dynamic problem, 105 Nonlinear free vibration FGM square plate, 109 isotropic square plate, 109 Nonlinear motion equations, 123 Nonlinear postbuckling load–shortening, 245–246 Nonlinear prebuckling deformations, 176

P Perturbation expansions boundary layer solutions, 185 Perturbation scheme buckling loads comparison, 163–165 description, 162 postbuckling load–deﬂection curves, 165 postbuckling load–shortening curves, 162–163 volume fraction index effect, 241 P=FGM plate, dynamic response temperature dependency, effect of, 132 volume fraction index effect, 132

263 Piezoelectric layer thermal conductivity, 121 thickness, 124 Plane stress constitutive equations, 11 Poincare method, 21 Poisson’s ratio, 125 Postbuckled isotropic rectangular plate, 137 Postbuckled sandwich plate, 138 fundamental frequencies, 142 nonlinear frequency ratio temperature change effects, 142 volume fraction index effect, 141, 143 temperature change effects, 140 thermal environments, 139 Postbuckling of FGM plates hybrid laminated plate (see Hybrid laminated plate) thermomechanical loads, 46 of FGM=SUS304=FGM sandwich plates material properties effect, 92 thermal effects on, 93 volume fraction index effect, 94–95 sandwich plates in thermal environments average end-shortening relationships, 86–87 boundary conditions, 86, 88 compressive postbuckling equilibrium path, 89 edges, simply supported, 85 FGM=SUS304=FGM plate, 92–96 immovability condition, 86 under uniform temperature, 89 Postbuckling equilibrium paths, 175, 190 Postbuckling load–deﬂection curves isotropic short cylindrical shell, 177, 178 isotropic thin plates, uniaxial compression, 58 shear deformable FGM plates, 57 Prebuckled sandwich plate nonlinear frequency ratio temperature changes, effects of, 142 volume fraction index, effects of, 141, 143

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264 plate stiffness, 138 temperature changes, effects of, 140 volume fraction index effects, 139, 142 Prebuckling deformation, 162

R Runge–Kutta iteration method, 118 Runge–Kutta iteration scheme, 108

S Sandwich plates conﬁguration of, 83 constituents of, 82 FGM face sheet, 83 material properties of, 83 postbuckling equilibrium states of, 135 postbuckling in thermal environments average end-shortening relationships, 86–87 boundary conditions, 86, 88 compressive postbuckling equilibrium path, 89 edges, simply supported, 85 FGM=SUS304=FGM plate, 92–96 immovability condition, 86 under uniform temperature, 89 temperature ﬁeld measurement, 83–85 Self-consistent models, 8–9 Shear deformable FGM plates general von Kármán-type equations bending moments, 15 equations of motion, 15–17 higher order moments, 15 thermal effect, 14 transverse shear forces, 15 load-bending moment curves, 57 load-end-shortening curves, 57 nonlinear bending under mechanical loads assumptions, 22 dimensionless quantities, 25 inplane behavior, 23–24 load-bending moment curves (see Load-bending moment curves)

Index load-central deﬂection curves (see Load-central deﬂection curves) nonlinear governing equations, 25–26 transverse static load, 23 two-step perturbation technique for, 27–30 unit end-shortening relationships, 24–25 postbuckling load–deﬂection, 57 Si3N4=SUS304 square plates (see Si3N4=SUS304 square plates) zirconia=aluminum square plate (see Zirconia=aluminum square plate) Shell end-shortening displacements, 155 Silicon nitride, 109 Simply supported shear deformable FGM plates nonlinear bending response, 27 thermal bending analysis, 22, 23 Si3N4=SUS304 cylindrical shell buckling shear stress comparison, 192, 193 buckling temperature, 204 imperfection sensitivity l*, 166, 169 postbuckling behavior temperature dependency, effect of, 166, 167 volume fraction index, effect of, 168 postbuckling load–shortening curves, 166 thermal postbuckling load–deﬂection curves clamped perfect=imperfect, 204 Si3N4=SUS304 moderately thick cylindrical shell postbuckling behavior temperature changes effect, 199 temperature dependency effect, 197 volume fraction, effect of, 198 Si3N4=SUS304 square plate, dynamic response, 120 Si3N4=SUS304 square plates load–bending moment curves for under in-plane boundary conditions, 41

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Index temperature-dependency, 40 volume fraction index effect, 39 material properties of, 57 thermal load–bending moment curve for under in-plane boundary conditions, 41 temperature-dependency, 40 volume fraction index effect, 39 thermal load-central deﬂection curve for effect of in-plane boundary conditions on, 41 effect of volume fraction index, 39 Si3N4=SUS304 thin cylindrical shell postbuckling behavior temperature changes effect, 196 temperature dependency effect, 194, 197 volume fraction index effect, 195 postbuckling load–shortening load–rotation curves, 194 torsional postbuckling, 197 Si3N4=US304 cylindrical shells buckling loads comparison, 178 postbuckling behavior temperature dependency effect, 180, 181 volume fraction index effect, 182 Stainless steel, 109 Steady-state heat conduction, 111 Steady-state heat transfer, 101 Stretching=bending coupling, 53

T Temperature-dependent (T-D), 148 Temperature distribution, 101 Temperature-independent, 148 Thermal buckling load hybrid laminated plates, 73, 77 Thermal buckling problem, 85 boundary conditions, 86, 88 Thermal conductivity piezoelectric layer, 121 Thermal load-deﬂection and loadbending moment curves FGM plate under heat conduction in-plane boundary conditions and, 41

265 temperature-dependency and, 40 volume fraction index and, 39 Thermally buckled plate, 134 Thermally buckled Si3N4=SUS304 square plate natural frequencies, comparisons of, 138 Thermal postbuckling behavior FGM=SUS304=FGM sandwich plates effects of volume fraction index on, 96 material properties effects, 95 hybrid laminated plates asymptotic solutions, 71–72 boundary conditions, 70–71 edges of plate, 67 material properties effect, 78–79 procedure to estimate, 72–73 temperature ﬁeld, 66 thermal load ratio effect, 80–81, 82 thermal postbuckling equilibrium path, 72 volume fraction index effect, 80, 81 Thermal postbuckling equilibrium path, 202 Thermal postbuckling load–deﬂection curves hybrid laminated plates, material properties effect, 78, 80 isotropic plates subjected to nonuniform parabolic temperature loading, 74 uniform temperature rise, 73 Thermomechanical loads, 21 Thermopiezoelectric loads, 122 T-ID. See Temperature-independent Titanium alloy, 109 Transverse shear strains, 150 Two-step perturbation technique nonlinear bending analysis by load–central deﬂection (see loadcentral deﬂection) small perturbation parameter, 27–28

V Voigt model, 5 von Kármán-type equations, 122, 133

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266 Y Young’s modulus, 101

Z Zirconia=aluminum square plate load–bending moment curves thermal environmental conditions, 31 transverse loading conditions, 34–35 uniform pressure conditions, 31, 33 volume fraction index, 32 load–bending moment curves for ‘‘immovable’’ in-plane boundary conditions, 33–34 ‘‘movable’’ in-plane boundary conditions, 33–34

Index load-central deﬂection curves for comparisons of, 31 in-plane boundary conditions effect, 34 temperature-dependency effect, 40 temperature rise effect, 33 uniform=sinusoidal load in thermal environments, 35 volume fraction index effect, 32 material properties of, 57 Zirconium oxide, 109 ZrO2=Ti-6Al-4V cylindrical hells buckling loads comparison, 179 thermal postbuckling load-deﬂection curves clamped perfect=imperfect, 204 ZrO2=Ti-6Al-4V square, plate dynamic response temperature ﬁeld effect, 119