Mechanics Today Volume 3 Edited by S. NEMAT-NASSER, Professor Department of Civil Engineering, The Technological Institute, Northwestern University, Evanston, Illinois Published by Pergamon Press on behalf of the AMERICAN ACADEMY OF MECHANICS PERGAMON PRESS INC. New York · Toronto · Oxford · Paris · Sydney · Frankfurt U.K. Pergamon Press Ltd., Headington Hill Hall, Oxford, England U. S. A. Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. CANADA Pergamon of Canada Ltd., P.O. Box 9600, Don Mills M3C 2T9, Ontario, Canada AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia FRANCE Pergamon Press SARL, 24 rue des Ecoles, 75240 Paris, Cedex 05, France WEST GERMANY Pergamon Press GmbH, 6242 Kronberg/Taunus, Pferdstrasse 1, Frankfurt-am-Main, West Germany Copyright © 1976 Pergamon Press Inc. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers First edition 1976 Library of Congress Cataloging in Publication Data Nemat-Nasser, S Mechanics today. (Pergamon mechanics today series) Vol. 1: 1972. "Published... on behalf of the American Academy of Mechanics." Includes bibliographical references. 1. Mechanics, Applied. I. American Academy of Mechanics. II. Title. TA350.N4 620.1 72-10430 ISBN 0-08-017246-6 (v. 1) ISBN 0-08-018113-9 (v. 2) ISBN 0-08-019882-1 (v. 3) Printed in Great Britain by A. Wheaton & Co. Exeter Contributors The number that follows each author's address refers to the page where his contribution begins. I A. Bedford, Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, Texas 78712, 1. D. S. Drumheller, Sandia Laboratories, Albuquerque, New Mexico 871 IS, 1. H. J. Sutherland, Sandia Laboratories, Albuquerque, New Mexico 871 IS, 1. II L. B. Freund, Division of Engineering, Brown University, Provi­ dence, Rhode Island 02912, 55 III Y. K. Lin, Department of Aeronautical and Astronautical Engineer- ing, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, 93. IV W. E. Olmstead, Department of Engineering Sciences, The Tech­ nological Institute, Northwestern University, Evanston, Illinois 60201, 125. A. K. Gautesen, Department of Mathematics, Clarkson College of Technology, Potsdam, New York 13676, 125. V P. R. Sethna, Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota SS4SS, 191. M. Balu Balachandra, Agbabian Associates, El Segundo, California 9024S, 191. VI Charles R. Steele, Department of Aeronautics and Astronautics, Stanford University, Stanford, California 9430S, 243. xi Preface The present volume of this series, Mechanics Today, follows the tradition that has been established by Volumes 1 and 2 and introduces the reader to contributions from some of the most active researchers in the fields of solid mechanics, fluid mechanics, and applied mathematics. The volume consists of six articles in areas of applied mechanics that are of current interest and have enjoyed a great deal of attention in the recent past. As in the first two volumes, each article begins with a discussion of funda­ mentals and proceeds with a presentation of analytical and experimental (where applicable) results. The subject matter is hence developed in such a manner that the article is useful to specialists, while at the same time it remains accessible to nonexperts with sufficient background. I wish to express my gratitude to Mrs. Erika Ivansons who has assisted with the editorial tasks. S. NEMAT-NASSER Wilmette, Illinois April 30,1975 xiii Contents of Volume 1 I Dynamic Effects in Brittle Fracture J. D. ACHENBACH II Qualitative Theory of the Ordinary Differential Equations of Nonlinear Elasticity STUART S. ANTMAN III Plastic Waves: Theory and Experiment R. J. CLIFTON IV Modern Continuum Thermodynamics MORTON E. GURTIN V General Variational Principles in Nonlinear and Linear Elasticity with Applications s. NEMAT-NASSER VI A Survey of Theory and Experiment in Viscometric Flows of Viscoelastic Liquids A. C. PIPKIN and R. i. TANNER VII Concepts in Elastic Structural Stability JOHN ROORDA xv Contents of Volume 2 I Theory of Creep and Shrinkage in Concrete Structures: A Précis of Recent Developments ZDENÉK P. BAZANT II On Nonequilibrium Thermodynamics of Continua s. NEMAT- NASSER III Mathematical Aspects of Finite-Element Approximations in Con­ tinuum Mechanics J. T. ODEN IV Nonlinear Geometrical Acoustics BRIAN R. SEYMOUR and MICHAEL P. MORTELL xvii Summary For the convenience of the reader, an abstract of each chapter of this volume is given below. I On Modeling the Dynamics of Composite Materials by A. Bedford, University of Texas at Austin, D. S. Drumheller and H. J. Sutherland, Sandia Laboratories During the past decade, a number of structured continuum theories have been developed for application to the mechanics of composite materials. In this article, a description of this approach is given and the types of results which have been obtained are surveyed in the context of one of the theories—the effective stiffness theory introduced by G. Herrmann, J. D. Achenbach, and C. T. Sun. In addition, a review is given of experimental methods which have provided data for verification of the theory. A review is first presented of theoretical and experimental evidence on the dynamical behavior of composites to illustrate the types of phenomena which a successful theory must model. A recent formulation of the effective stiffness theory is then described in detail, with equations of motion and boundary conditions presented for the case of plain strain motion. Results on steady-state and transient wave propagation which have been obtained with the theory are then reviewed with emphasis on those that have been compared with experimental and other theoretical results. The methods which have been used to obtain experimental data are then described, followed by comments on the possible future course of the theoretical development. II The Analysis of Elastodynamic Crack Tip Stress Fields by L. B. Freund, Brown University Because of loading conditions or material characteristics, there are numerous fracture mechanics problems in which the inertia of the xix xx Summary material must be taken into account. The purpose of this article is to summarize some of the analytical results which have recently been obtained in an effort to improve understanding of such elastodynamic fracture processes. It is assumed throughout that the state of the deformation at the crack tip is plane strain, and both a stationary crack in a dynamically loaded body and a rapidly extending crack in a stressed body are considered. After a brief review of the basic equations of linear elastodynamics, the fact that the crack tip stress field is completely characterized by the time-dependent stress intensity factor is demonstrated by establishing the universal spatial dependence of the near tip field. Next, a study of energy variations with crack size leads to a path-independent integral, which is of use in solving stationary crack problems, and to an expression for the energy release rate for an extending crack, which is subsequently used to establish uniqueness of running crack solutions. Finally, three specific problem cases, in which the determination of the dynamic stress intensity factor is the central objective, are considered in some detail. These are: (a) the analysis of stationary cracks by means of the weight function method; (b) the analysis of crack extension at a nonuniform rate by an inverse method; and (c) the analysis of stationary or extending cracks by the method of continuously distributed moving dislocations. Considera­ tion of each problem class is concluded with a discussion of a typical example. Ill Random Vibration of Periodic and Almost Periodic Structures by Y. K. Lin, University of Illinois at Urbana-Champaign A periodic structure is one which is constructed by connecting together identical units. Due to close clustering of the natural frequencies of such structures, the usual normal mode approach is inefficient for calculating the response statistics. In the present article matrix difference equations are used to formulate such problems, taking advantage of the identical construction of the interconnecting units. The general procedure is to determine the frequency response functions and then use them in spectral analyses. The case of discrete periodic systems is discussed first, followed by continuous systems. Since perfect periodicity never exists in practice, the effect of small random deviations from periodicity is investigated. Some numerical results are presented graphically as spectral densities of the response under the excitation of weakly stationary or homogeneous random processes. Summary xxi IV Integral Representations and the Oseen Flow Problem by W. E. Olmstead, Northwestern University, and A. K. Gautesen, Clarkson College of Technology The Oseen equations are an appropriate approximation of the Navier-Stokes equations for many problems in low Reynolds number fluid mechanics. Although linear, the Oseen equations still pose a rather formidable problem for solution. This article develops a basic approach to the Oseen problem through the use of integral representations of the velocity and pressure fields. With this approach, the solution of a given problem is reduced to that of solving an integral equation for the unknown stress at the fluid boundaries. In the case of flow past a fixed body, this leads directly to the determination of the forces exerted by the fluid on the body. Many example problems are treated where known results on integral equations and related variational methods can be utilized. These include flows past plates and circular arcs as well as injection-suction and free surface problems. V On Nonlinear Gyroscopic Systems by P. R. Sethna, University of Minnesota, and M. Balu Balachandra, Agbabian Associates This study is concerned with the analysis of nonlinear gyroscopic systems. The governing equations of motion for a nonlinear gyroscopic system are derived in the case when the system has prescribed imposed motions and also in the case when the system has conserved momenta. The study is done for two basic cases. In one case the motions are studied when they are in the neighborhood of stable steady motions and in the other case they are studied when the imposed motions are fast or alternatively when the conserved momenta are large. In this latter case no restrictions are imposed on the magnitude of the displacements or velocities. The method of analysis has asymptotic validity and is based on the Method of Averaging and its generalizations. Motions of high-order nongyroscopic systems are discussed as special cases. VI Application of the WKB Method in Solid Mechanics by Charles R. Steele, Stanford University In the study of static deformation, vibration, wave propagation, and instability of elastic bodies, equations with variable coefficients are often encountered which may be effectively handled by some variation of the so-called WKB perturbation method. The well-known first approximation xxii Summary solution for the second-order equation is reviewed. As indicated by examples of eigenvalue and pulse propagation problems, the first approximation solution generally is better than one would expect for equations with coefficients which are not too "smooth". The variational method of Whitham provides an elegant, simple derivation of the first approximation solution for conservative systems of higher order. In a perturbation method, it is natural to seek successive corrections, since improvement in both numerical accuracy and in the understanding of the limitation on the first approximation is to be gained. Included are the formal asymptotic expansion for the matrix equation of nth order and the convergent expansion obtained by Keller and Keller. Judging by a few examples for the second-order equation, the first correction term of the convergent series is actually easier to use than that of the formal expansion and provides a substantial gain in accuracy. A novel feature of equations with variable coefficients without counter­ part in homogeneous systems are "transition points". At these points the roots of the eikonal equation of the first approximation WKB solution coalesce, invalidating the approximation. The uniformly valid solution for the second-order equation due to Langer is extended to apply to the nth order system with two roots which coalesce. The solution for the case of four roots which coalesce is also given. This frequently occurs in beam and shell problems. For a final example, a simple formula is obtained for the calculation of the lowest natural frequency of asymmetric vibration of a thin conical shell with constrained ends.