Tensor Analysis and Continuum Mechanics Wilhelm Flügge Springer-Verlag Berlin Heidelberg GmbH 1972 Dr.-In g. Wilhelm Fltigge Professor of Applied Mechanics, emeritus Stanford UiJiversity With 58 Figures ISBN 978-3-642-88384-2 ISBN 978-3-642-88382-8 (eBook) DOI 10.1007/978-3-642-88382-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con cerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under §54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.© Springer-Verlag Berlin Heidelberg 1972. Softcover reprint of the hardcover 1st edition 1972 Library of Congress Catalog Card Number 74-183541 Preface Through several centuries there has been a lively interaction between mathematics and mechanics. On the one side, mechanics has used mathemat ics to formulate the basic laws and to apply them to a host of problems that call for the quantitative prediction of the consequences of some action. On the other side, the needs of mechanics have stimulated the development of mathematical concepts. Differential calculus grew out of the needs of Newtonian dynamics; vector algebra was developed as a means .to describe force systems; vector analysis, to study velocity fields and force fields; and the calcul~s of variations has evolved from the energy principles of mechan ics. In recent times the theory of tensors has attracted the attention of the mechanics people. Its very name indicates its origin in the theory of elasticity. For a long time little use has been made of it in this area, but in the last decade its usefulness in the mechanics of continuous media has been widely recognized. While the undergraduate textbook literature in this country was becoming "vectorized" (lagging almost half a century behind the development in Europe), books dealing with various aspects of continuum mechanics took to tensors like fish to water. Since many authors were not sure whether their readers were sufficiently familiar with tensors~ they either added' a chapter on tensors or wrote a separate book on the subject. Tensor analysis has undergone notable changes in this process, especially in notations and nomenclature, but also in a shift of emphasis and in the establishment of a cross connection to the Gibbs type of vector analysis (the "boldface vectors "). Many of the recent books on continuum mechanics are only" tensorized " to the extent that they use cartesian tensor notation as a convenient iv Preface shorthand for writing equations. This is a rather harmless use of tensors. The general, noncartesian tensor is a much sharper thinking tool and, like other sharp tools, can be very beneficial and very dangerous, depending on how it is used. Much nonsense can be hidden behind a cloud of tensor symbols and much light can be shed upon a difficult subject. The more thoroughly the new generation of engineers learns to understand and to use tensors, ,the more useful they will be. This book has been written with the intent to promote such understanding. It has grown out of a graduate Course that teaches tensor analysis against the background of its application in mechanics. As soon as each mathematical concept has been developed, it is interpreted in mechanical terms and its use in continuum mechanics is shown. Thus, chapters on mathematics a~d on mechanics alternate, and it is hoped that this will bring lofty tbeory down to earth and help the engineer to understand the creations of abstract thinking in terms of familiar objects. Mastery of a mathematical tool cannot be acquired by just reading about it-it needs practice. In order that the reader may get started on his way to practice, problems have been attached to most chapters. The reader is encouraged to solve them and then to proceed further, and to apply what he has learned to his own problems. This is what the author did when, several decades ago, he was first confronted with the need of penetrating the thicket of tensor books of that era. The author wishes to express his thanks to Dr. William Prager for critically reading the manuscript, and to Dr. Tsuneyoshi Nakamura, who persuaded him to give a series of lectures at Kyoto University. The preparation of these lectures on general sheU theory gave the final push toward starting work on this book. Stanford, California W.F. Contents CHAPTER 1. Vectors and Tensors 1 1.1. Dot Product, Vector Components 1.2. Base Vectors, Metric Tensor 7 1.3. Coordinate Transformation 12 1.4. Tensors 15 Problems 21 References 21 CHAPTER 2. The Strain Tensor 23 Problem 28 References 28 CHAPTER 3. The Cross Product 29 3.1. Permutation Tensor 29 3.2. Cross Product 36 Problems 43 CHAPTER 4. Stress 44 4.1. Stress Tensor 44 4.2. Constitutive Equations SO 4.3. Plasticity 60 Problem 65 References 65 vi Contents CHAPTER 5. Derivatives and Integrals 66 5.1. Christoffel Symbols 66 5.2. Covariant Derivative 68 5.3. Divergence and Curl 74 5.4. The Integral Theorems of Stokes and Gauss 76 Problems 83 References 84 CHAPTER 6. The Fundamental Equations of Continuum Mechanics 85 6.1. Kinematic Relations 85 6.2. Condition of Equilibrium and Equation of Motion 87 6.3. Fundamental Equation of the Theory of Elasticity 89 6.4. Flow of Viscous Fluids 93 6.5. Seepage Flow 99 Problems 104 References 104 CHAPTER 7. Special Problems of Elasticity 105 7.1. Plane Strain 105 7.2. Plane Stress 112 7.3. Generalized Plane Strain 113 7.4. Torsion 116 7.5. Plates 123 Problem 130 References 130 CHAPTER 8. Geometry of Curved Surfaces 131 8.1. General Considerations 131 8.2. Metric and Curv ... ture 133 8.3. Covariant Derivatiz7e 138 Problems 141 CHAPTER 9. Theory of Shells 143 9.1. Shell Geometry 143 9.2. Kinematics of Deformation 147 9.3. Stress Resultants and Equilibrium 153 9.4. Elastic Law 161 Problems 163 References 163 CHAPTER 10. Elastic Stability 165 References 171 Contents vii CHAPTER 11. Principal Axes and Invariants 172 11.1. Unsymmetric Tensor 173 11.2. Tensor of Stress and Strain 176 11.3. Curvature 179 11.4. Vectors 180 Problem 181 CHAPTER 12. Compilation of Tensor Formulas 182 12.1. Mathematical Formulas 182 12.2. Mechanical Formulas 187 CHAPTER 13. F-ormulas for Special Coordinate Systems 193 13.1. Plane Polar Coordinates 193 13.2. Plane Elliptic-Hyperbolic Coordinates 194 13.3. Plane Bipolar Coordinates 194 13.4. Skew Rectilinear Coordinates 196 13.5. Cylindrical Coordinates 196 13.6. Spherical Coordinates 197 13.7. Skew Circular Cone 198 13.8. Right Circular Cone 199 13.9. Hyperbolic Paraboloid 200 Bibliography 202 Index 204 CHAPTER 1 Vectors and Tensors IT IS ASSUMED THAT the reader is familiar with the representation of vectors by arrows, with their addition and their resolution into components, i.e. with the vector parallelogram and its extension to three dimensions. We also assume familiarity with the dot product and later (p. 36) with the cross product. Vectors SUbjected to this special kind of algebra will be called Gibbs type vectors and will be denoted by boldface letters. In this and the following sections the readf:r will learn a completely different means of describing the same physical quantities, called tensor algebra. Each of the two competingformulations has its advantages and its drawbacks. The Gibbs form of vector algebra is independent of a coordinate system, appeals strongly to visualization and leads easily into graphical methods, while tensor algebra is tied to coordinates, is abstract and very formal. This puts the tensor formulation of physical problems at a clear disadvantage as long as one deals with simple objects, but makr.s it a powerful tool in situa tions too complicated to permit visualization. The Gibbs formalism can be extended to physical quantities more complicated than a vector (moments of inertia, stress, strain), but this extension is _rather cumbersome and rarely used. On the other hand, in tensor algebra the vector appears as a special case of a more general concept, which includes stress and inertia tensors but is easily extended beyond them. 1.1. Dot Product, Vector Components In a cartesian coordinate system x, y, z (Figure 1.1) we define a reference frame of unit vectors il<; i" i: along the coordinate axes and with their help a force vector 2 Vectors and Tensors [Ch.l p =P)" +P)y +Pziz (l.1a) and a displacement vector u = u)" + uri,)' + uziz• (1.1 b) These formulas include the well-known definition of the addition of vectors by the parallelogram rule. In mechanics the work W done by the force P during a displacement u is defined as the product of the absolute values P and u of the two vectors and of the cosine of the angle p between them: W = Pu cos p. Thi.s may be interpreted as the product of the force and the projection of u on the direction of P or as the product of the displacement and the projection of the force on u. It is commonly written as the dot product of the two vectors: w = P . u = u . P = Pu cos p. (1.2) This equation represents the definition of the dot product and may be applied to any two vectors. Since the projection of a vector 11 = V + w on v the direction of P is equal to the sum of the projections of and w, it is evident that the dot product has the distributive property: p . (v + w) = P . v + p . w. When anyone of the unit vectors i", i)/, i is dot-multiplied with itself, the z p angle of (l.2) is zero, hence u FJGUlU, 1,1 Vectors in cartesian "oordinates.