Stress and Strain: Basic Concepts of Continuum Mechanics for Geologists PDF
Preview Stress and Strain: Basic Concepts of Continuum Mechanics for Geologists
STRESS AND STRAIN STRESS AND STRAIN Basic Concepts of Continuum Mechanics for Geologists W. D. MEANS I Springer-Verlag New York Heidelberg Berlin 1976 W. D. Means Department of Geological Sciences State University of New York at Albany Albany, New York Library of Congress Cataloging in Publication Data Means, Winthrop Dickinson. Stress and strain. Includes bibliographical references and index. 1. Rock deformation. 2. Rock mechanics. 3. Continuum mechanics. I. Title. QE604.M4 551.8 75-34468 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1976 by Springer-Verlag New York Inc. ISBN-13: 978-0-387-07556-3 e-ISBN-13: 978-1-4613-9371-9 DOl: 10.1007/978-1-4613-9371-9 To Anna Preface This is an elementary book on stress and strain theory for geologists. It is written in the belief that a sound introduction to the mechanics of continu ous bodies is essential for students of structural geology and tectonics, just as a sound introduction to physical chemistry is necessary for students of petrology. This view is shared by most specialists in structural geology, but it is not yet reflected in typical geology curricula. Undergraduates are still traditionally given just a few lectures on mechanical fundamentals, and there is rarely any systematic lecturing on this subject at the graduate level. The result is that many students interested in structure and tectonics finish their formal train ing without being able to understand or contribute to modem literature on rocks as mechanical systems. The long-term remedy for this is to introduce courses in continuum mechanics and material behavior as routine parts of the undergraduate curriculum. These subjects are difficult, but no more so than optical mineralogy or thermo dynamics or other rigorous subjects customarily studied by undergraduates. The short-term remedy is to provide books suitable for independ ent study by those students and working geologists alike who wish to improve their understanding of mechanical topics relevant to geology. This book is intended to meet the short-term need with respect to stress and strain, two elementary yet challenging concepts of continuum mechanics. It is written primarily as a self-instruction manual, but it can also serve as text for a semester lecture course, or to supplement the main text in a struc tural geology course. The mathematical theory is developed here in very elementary and extended fashion. The treat ment will seem tedious in places to those already vii Preface familiar with the subject, but it should be helpful to beginners, especially those with meager back grounds in physics and applied mathematics. To make the material as digestible as possible, there are twenty-seven short chapters, and each chapter is followed by a set of solved problems. It is hoped that early study of this introductory book will complement and facilitate use of the generally more advanced and comprehensive texts of Jaeger (1969) and Ramsay (1967) and the problem oriented books of Johnson (1970) and Hubbert (1972). The treatment here is most akin to, and is partly inspired by, chapters by W. M. Chapple and E. G. Bombolakis in the Report of the 1967 Rock Mechanics Seminar, edited by Riecker (1968). The emphasis is on the concepts of stress and strain rather than on the solution of particular geological problems. Readers who want to read a shortened version of the text, omitting some of the parts on stress and strain tensors, are advised to skip Chapters 11, 20,21, and 22. Albany, New York W. D. Means August, 1975 viii Acknowledgments I am particularly indebted to Brian Bayly (Renn selaer Polytechnic Institute, USA) and Nick Gay (University of Witwatersrand, South Africa). These two colleagues read the first draft and suggested many ways to improve it. I have also been greatly helped by students, especially Bob Findlay (Auckland University, New Zealand) who systematically went through the second draft and pointed out places where the text or figures were confusing. Other students who have given special help in this way include Anne Wright, Julie Milligan and Terry Crippen (Auckland University) and Bruce Nisbet and Suzanne O'Connell (State University of New York at Albany). In addition I acknowledge the major contributions of Bruce Hobbs (Monash University, Australia) and Paul Williams (Rijksuniversiteit, Leiden, The Netherlands). Numerous discussions with these two associates over a period of years have done much to upgrade my own thinking about stress and strain. Despite all this assistance I have probably allowed some errors to remain in the text; I shall be glad to hear from readers who find them. I thank Miss Terri VanDerwerken for her patient typing of difficult material and Professor R. N. Brothers and the staff of the Geology Department, Auckland University, for congenial surroundings during sabbatical leave in 1975 when the main work on the book was finished. ix Contents Part I 1 Material constitution of rocks 2 Introductory 2 Mechanical state 6 3 Change in mechanical state 13 4 Mechanical significance of structure 20 Part II 5 Classes of forces 36 Forces in Rocks 6 Stress on a plane 42 7 The stress ellipsoid, I 53 8 The stress ellipsoid, II 61 9 Mohr circle for stress 70 10 Tensor components of stress 85 11 Cauchy's formula, transformation of tensor components 100 12 Stress fields 111 13 Stress history 122 Part III 14 Distortion and deformation, measures of Deformation of distortion 130 Rocks 15 The strain ellipsoid 139 16 Mohr circle for infinitesimal strain 151 17 Mohr circle for finite strain 159 18 Displacement and deformation gradients 168 19 Tensor components of infinitesimal strain, I 181 20 Tensor components of infinitesimal strain, II 188 21 Tensor components of finite strain, I 196 22 Tensor components of finite strain, II 203 23 Strain fields 215 24 Strain history 224 Part IV 25 Hookean behavior 240 Topics Involving 26 Newtonian behavior 254 Forces and 27 Energy consumed in deformation 263 Deformation Solutions to Problems 275 Index 335 xi I INTRODUCTORY Part Mechanics is concerned with forces and motions in any material system. Some knowledge of mechan ics is valuable in geology because many observable properties of rock bodies reflect aspects of the forces or motions the bodies have experienced. Such properties include structures, microstructures, mineral assemblages, lithic sequences in sedimentary basins, and even faunal sequences. Most of this book deals, however, not with mechanical inter pretation of rocks themselves but with certain elementary concepts of mechanics that are best explained with reference to an idealized homogene ous and continuous material. In Chapter 1 the dif ferences and similarities between this imaginary material and real rock are discussed. In Chapters 2 and 3 an important distinction is introduced between viewing the mechanical state of a rock body at an instant and comparing its state at two different instants. In Chapter 4 there is an overview of the mechanical significance of geo logical features. 1 1 Material Constitution of Rocks Homogeneity and We start by defining four classes of material. continuity of A material is homogeneous if the properties of materials one piece are the same as the properties of any other piece. A material that does not fit this definition is said to be inhomogeneous. A material is continuous if every subvolume within it is occupied by the material and if its properties vary smoothly from one point to another. A materia). that does not have these characteristics, in which properties such as density and strength vary a'bruptly across internal surfaces, is said to be discontinuous. When we apply these definitions to granular, crystalline materials like rocks or soils, metals or ceramics, we find that all these materials are inhomogeneous and discontinuous. Even the most uniform and solid-looking cube of quartzite, for example, will contain minute openings along grain boundaries, and any two parts of the cube will have slightly different densities. Real materials are thus never perfectly homogeneous or perfectly continuous. The concepts of perfect homogeneity and perfect continuity are nevertheless useful because they define the properties of the idealized materials dealt with in much of the mathematical theory of mechanical behavior. These concepts also permit us to classify material as approximately homo geneous or approximately continuous if we specify permissible departures from the ideal. This is illustrated for honiogeneity in Figure 1.1. Notice that to call a region of real material homogeneous in this looser, or "statistical," sense, we have to specify a scale on which the region is sampled i and also the particular properties with respect to which we are examining it. 2
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