Advances in Applied Mechanics Volume 21 Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERM” HILL RODNEY L. HOWARTH T. Y. Wu Contributors to Volume 21 B. L. N. KENNETT L. J . WALPOLE J . R. WILLIS AD VANCES I N APPLIED MECHANICS Edited by Chia-Shun Yih DEPARTMENT OF MECHANICAL ENGINEERING AND APPLIED MECHANICS THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN VOLUME 21 1981 ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Toronto Sydney San Francisco COPYRIGHT @ 1981, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STOR4GE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. ( L O N D O N ) LTD. 2 4 / 2 8 Oval Road, London N W 1 7 D X LIBRARY OF CONGRESS CATALOG CARD NUMBER:48-8503 ISBN 0-12-002021-1 PRINTED IN THE UNITED STATES O F AMERICA 81 82 83 84 9 8 7 6 5 4 3 2 1 Contents vii LISTOF CONTRIBUTORS Variational and Related Methods for the Overall Properties of Composites J . R . Willis I. Introduction 11. 111. IV. V. VI. VII. VIII. Preliminary Definitions Methods Related to the Classical Variational Principles Methods Related to the Hashin-Shtrikman Variational Principle Self-consistent Estimates Generalizations Problems Which Lack Convergence Wave Propagation IX . Recent Developments References 2 3 13 23 42 47 55 64 74 74 Elastic Wave Propagation in Stratified Media B. L. N . Kennett I. Introduction 11. Elastic Waves in Stratified Regions 111. Reflection and Transmission of Elastic Waves IV. Half-Space Response in Terms of Reflection Matrices V. Inversion of the Transforms VI. Conclusion References 80 83 100 127 152 163 164 Elastic Behavior of Composite Materials: Theoretical Foundations L . J . Walpole 1. Introduction 11. Preliminary Analysis of Tensors and Elastic Behavior V 169 171 vi 111. The Elastic Field of an Inclusion IV. The Elastic Field of a Composite Body V. The Overall Elastic Behavior of a Composite Body References Contents 188 202 208 236 AUTHORINDEX 243 SUBJECT INDEX 241 List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin. B. L. N. KENNETT,Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, England (79) L. J. WALPOLE, School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ, England (169) J. R. WILUS,School of Mathematics, University of Bath, Bath BA2 7AY, England (1) vii This Page Intentionally Left Blank ADVANCES IN APPLIED MECHANICS. VOLUME 21 Variational and Related Methods for the Overall Properties of Composites J . R . WILLIS School of Mathematics University of Bath Bath. England I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Types of Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Definitions of Overall Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 3 1 I11. Methods Related to the Classical Variational Principles . . . . . . . . . . . . . . . A. Small Variations in Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . .