ELASTICITY IN ENGINEERING MECHANICS Elasticity in Engineering Mechanics, Third Edition Arthur P. Boresi, Ken P. Chong and James D. Lee Copyright © 2011 John Wiley & Sons, Inc. ELASTICITY IN ENGINEERING MECHANICS Third Edition ARTHUR P. BORESI Professor Emeritus University of Illinois, Urbana, Illinois and University of Wyoming, Laramie, Wyoming KEN P. CHONG Associate National Institute of Standards and Technology, Gaithersburg, Maryland and Professor Department of Mechanical and Aerospace Engineering George Washington University, Washington, D.C. JAMES D. LEE Professor Department of Mechanical and Aerospace Engineering George Washington University, Washington, D.C. JOHN WILEY & SONS, INC. This book is printed on acid-free paper. Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Boresi, Arthur P. (Arthur Peter), 1924- Elasticity in engineering mechanics / Arthur P. Boresi, Ken P. Chong and James D. Lee. – 3rd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-470-40255-9 (hardback : acid-free paper); ISBN 978-0-470-88036-4 (ebk); ISBN 978-0-470-88037-1 (ebk); ISBN 978-0-470-88038-8 (ebk); ISBN 978-0-470-95000-5 (ebk); ISBN 978-0-470-95156-9 (ebk); ISBN 978-0-470-95173-6 (ebk) 1. Elasticity. 2. Strength of materials. I. Chong, K. P. (Ken Pin), 1942- II. Lee, J. D. (James D.) III. Title. TA418.B667 2011 620.1  1232–dc22 2010030995 Printed in the United States of America 10987654321 CONTENTS Preface xvii CHAPTER 1 INTRODUCTORY CONCEPTS AND MATHEMATICS 1 Part I Introduction 1 1-1 Trends and Scopes 1 1-2 Theory of Elasticity 7 1-3 Numerical Stress Analysis 8 1-4 General Solution of the Elasticity Problem 9 1-5 Experimental Stress Analysis 9 1-6 Boundary Value Problems of Elasticity 10 Part II Preliminary Concepts 11 1-7 Brief Summary of Vector Algebra 12 1-8 Scalar Point Functions 16 1-9 Vector Fields 18 1-10 Differentiation of Vectors 19 1-11 Differentiation of a Scalar Field 21 1-12 Differentiation of a Vector Field 21 1-13 Curl of a Vector Field 22 1-14 Eulerian Continuity Equation for Fluids 22 v vi CONTENTS 1-15 Divergence Theorem 25 1-16 Divergence Theorem in Two Dimensions 27 1-17 Line and Surface Integrals (Application of Scalar Product) 28 1-18 Stokes’s Theorem 29 1-19 Exact Differential 30 1-20 Orthogonal Curvilinear Coordiantes in Three-Dimensional Space 31 1-21 Expression for Differential Length in Orthogonal Curvilinear Coordinates 32 1-22 Gradient and Laplacian in Orthogonal Curvilinear Coordinates 33 Part III Elements of Tensor Algebra 36 1-23 Index Notation: Summation Convention 36 1-24 Transformation of Tensors under Rotation of Rectangular Cartesian Coordinate System 40 1-25 Symmetric and Antisymmetric Parts of a Tensor 46 1-26 Symbols δ ij and  ijk (the Kronecker Delta and the Alternating Tensor) 47 1-27 Homogeneous Quadratic Forms 49 1-28 Elementary Matrix Algebra 52 1-29 Some Topics in the Calculus of Variations 56 References 60 Bibliography 63 CHAPTER 2 THEORY OF DEFORMATION 65 2-1 Deformable, Continuous Media 65 2-2 Rigid-Body Displacements 66 2-3 Deformation of a Continuous Region. Material Variables. Spatial Variables 68 2-4 Restrictions on Continuous Deformation of a Deformable Medium 71 Problem Set 2-4 75 2-5 Gradient of the Displacement Vector. Tensor Quantity 76 CONTENTS vii 2-6 Extension of an Infinitesimal Line Element 78 Problem Set 2-6 85 2-7 Physical Significance of  ii .Strain Definitions 86 2-8 Final Direction of Line Element. Definition of Shearing Strain. Physical Significance of  ij (i = j) 89 Problem Set 2-8 94 2-9 Tensor Character of  αβ . Strain Tensor 94 2-10 Reciprocal Ellipsoid. Principal Strains. Strain Invariants 96 2-11 Determination of Principal Strains. Principal Axes 100 Problem Set 2-11 106 2-12 Determination of Strain Invariants. Volumetric Strain 108 2-13 Rotation of a Volume Element. Relation to Displacement Gradients 113 Problem Set 2-13 116 2-14 Homogeneous Deformation 118 2-15 Theory of Small Strains and Small Angles of Rotation 121 Problem Set 2-15 130 2-16 Compatibility Conditions of the Classical Theory of Small Displacements 132 Problem Set 2-16 137 2-17 Additional Conditions Imposed by Continuity 138 2-18 Kinematics of Deformable Media 140 Problem Set 2-18 146 Appendix 2A Strain–Displacement Relations in Orthogonal Curvilinear Coordinates 146 2A-1 Geometrical Preliminaries 146 2A-2 Strain–Displacement Relations 148 Appendix 2B Derivation of Strain–Displacement Relations for Special Coordinates by Cartesian Methods 151 2B-1 Cylindrical Coordinates 151 2B-2 Oblique Straight-Line Coordinates 153 viii CONTENTS Appendix 2C Strain–Displacement Relations in General Coordinates 155 2C-1 Euclidean Metric Tensor 155 2C-2 Strain Tensors 157 References 159 Bibliography 160 CHAPTER 3 THEORY OF STRESS 161 3-1 Definition of Stress 161 3-2 Stress Notation 164 3-3 Summation of Moments. Stress at a Point. Stress on an Oblique Plane 166 Problem Set 3-3 171 3-4 Tensor Character of Stress. Transformation of Stress Components under Rotation of Coordinate Axes 175 Problem Set 3-4 179 3-5 Principal Stresses. Stress Invariants. Extreme Values 179 Problem Set 3-5 183 3-6 Mean and Deviator Stress Tensors. Octahedral Stress 184 Problem Set 3-6 189 3-7 Approximations of Plane Stress. Mohr’s Circles in Two and Three Dimensions 193 Problem Set 3-7 200 3-8 Differential Equations of Motion of a Deformable Body Relative to Spatial Coordinates 201 Problem Set 3-8 205 Appendix 3A Differential Equations of Equilibrium in Curvilinear Spatial Coordinates 207 3A-1 Differential Equations of Equilibrium in Orthogonal Curvilinear Spatial Coordinates 207 3A-2 Specialization of Equations of Equilibrium 208 3A-3 Differential Equations of Equilibrium in General Spatial Coordinates 210 CONTENTS ix Appendix 3B Equations of Equilibrium Including Couple Stress and Body Couple 211 Appendix 3C Reduction of Differential Equations of Motion for Small-Displacement Theory 214 3C-1 Material Derivative. Material Derivative of a Volume Integral 214 3C-2 Differential Equations of Equilibrium Relative to Material Coordinates 218 References 224 Bibliography 225 CHAPTER 4 THREE-DIMENSIONAL EQUATIONS OF ELASTICITY 226 4-1 Elastic and Nonelastic Response of a Solid 226 4-2 Intrinsic Energy Density Function (Adiabatic Process) 230 4-3 Relation of Stress Components to Strain Energy Density Function 232 Problem Set 4-3 240 4-4 Generalized Hooke’s Law 241 Problem Set 4-4 255 4-5 Isotropic Media. Homogeneous Media 255 4-6 Strain Energy Density for Elastic Isotropic Medium 256 Problem Set 4-6 262 4-7 Special States of Stress 266 Problem Set 4-7 268 4-8 Equations of Thermoelasticity 269 4-9 Differential Equation of Heat Conduction 270 4-10 Elementary Approach to Thermal-Stress Problem in One and Two Variables 272 Problem 276 4-11 Stress–Strain–Temperature Relations 276 Problem Set 4-11 283 4-12 Thermoelastic Equations in Terms of Displacement 285 4-13 Spherically Symmetrical Stress Distribution (The Sphere) 294 Problem Set 4-13 299 x CONTENTS 4-14 Thermoelastic Compatibility Equations in Terms of Components of Stress and Temperature. Beltrami–Michell Relations 299 Problem Set 4-14 304 4-15 Boundary Conditions 305 Problem Set 4-15 310 4-16 Uniqueness Theorem for Equilibrium Problem of Elasticity 311 4-17 Equations of Elasticity in Terms of Displacement Components 314 Problem Set 4-17 316 4-18 Elementary Three-Dimensional Problems of Elasticity. Semi-Inverse Method 317 Problem Set 4-18 323 4-19 Torsion of Shaft with Constant Circular Cross Section 327 Problem Set 4-19 331 4-20 Energy Principles in Elasticity 332 4-21 Principle of Virtual Work 333 Problem Set 4-21 338 4-22 Principle of Virtual Stress (Castigliano’s Theorem) 339 4-23 Mixed Virtual Stress–Virtual Strain Principles (Reissner’s Theorem) 342 Appendix 4A Application of the Principle of Virtual Work to a Deformable Medium (Navier–Stokes Equations) 343 Appendix 4B Nonlinear Constitutive Relationships 345 4B-1 Variable Stress–Strain Coefficients 346 4B-2 Higher-Order Relations 346 4B-3 Hypoelastic Formulations 346 4B-4 Summary 347 Appendix 4C Micromorphic Theory 347 4C-1 Introduction 347 4C-2 Balance Laws of Micromorphic Theory 350 4C-3 Constitutive Equations of Micromorphic Elastic Solid 351 CONTENTS xi Appendix 4D Atomistic Field Theory 352 4D-1 Introduction 353 4D-2 Phase-Space and Physical-Space Descriptions 353 4D-3 Definitions of Atomistic Quantities in Physical Space 355 4D-4 Conservation Equations 357 References 359 Bibliography 364 CHAPTER 5 PLANE THEORY OF ELASTICITY IN RECTANGULAR CARTESIAN COORDINATES 365 5-1 Plane Strain 365 Problem Set 5-1 370 5-2 Generalized Plane Stress 371 Problem Set 5-2 376 5-3 Compatibility Equation in Terms of Stress Components 377 Problem Set 5-3 382 5-4 Airy Stress Function 383 Problem Set 5-4 392 5-5 Airy Stress Function in Terms of Harmonic Functions 399 5-6 Displacement Components for Plane Elasticity 401 Problem Set 5-6 404 5-7 Polynomial Solutions of Two-Dimensional Problems in Rectangular Cartesian Coordinates 408 Problem Set 5-7 411 5-8 Plane Elasticity in Terms of Displacement Components 415 Problem Set 5-8 416 5-9 Plane Elasticity Relative to Oblique Coordinate Axes 416 Appendix 5A Plane Elasticity with Couple Stresses 420 5A-1 Introduction 420 5A-2 Equations of Equilibrium 421 [...]... University of Illinois – Urbana) Although much significant progress has been made in the field of bioscience and technology, especially in biomechanics, there exist many open problems related to elasticity, including molecular and cell biomechanics, biomechanics of development, biomechanics of growth and remodeling, injury biomechanics and rehabilitation, functional tissue engineering, muscle mechanics and... solving A course for first-year graduate students in civil and mechanical engineering and related engineering fields can include Chapters 1 through 6, with selected materials from the appendixes and/or Chapters 7 and 8 A follow-up graduate course can include most of the appendix material in Chapters 2 to 6, and the topics in Chapters 7 and 8, with specialized topics of interest for further study by individual... Continuum mechanics concepts such as couple stress and body couple are introduced into the theory of stress in the appendices of Chapters 3, 5, and 6 These effects are introduced into the theory in a direct way and present no particular problem The notations of stress and of strain are based on the concept of a continuum, that is, a continuous distribution of matter in the region (space) of interest In. .. in Terms of the Galerkin Vector F Problem Set 8-5 8-6 The Galerkin Vector: A Solution of the Equilibrium Equations of Elasticity Problem Set 8-6 8-7 The Galerkin Vector kZ and Love’s Strain Function for Solids of Revolution Problem Set 8-7 8-8 Kelvin’s Problem: Single Force Applied in the Interior of an In nitely Extended Solid Problem Set 8-8 8-9 The Twinned Gradient and Its Application to Determine... optimization of large engineering systems (Atrek et al., 1984; Zienkiewicz and Taylor, 2005; Kirsch, 1993; Tsompanakis et al., 2008) such as the space shuttle In addition, computers have played a powerful role Elasticity in Engineering Mechanics, Third Edition Copyright © 2011 John Wiley & Sons, Inc Arthur P Boresi, Ken P Chong and James D Lee 1 2 INTRODUCTORY CONCEPTS AND MATHEMATICS in the fields of computer-aided... nonlinear theory of elasticity (Green and Adkins, 1970) However, if the relationship of the stress and the deformation is linear, the material is said to be linearly elastic, and the corresponding theory is called the linear theory of elasticity The major part of this book treats the linear theory of elasticity Although ad hoc in form, this theory of elasticity plays an important conceptual role in. .. Plane Elasticity Boundary Value Problems in Complex Form 5B-7 Note on Conformal Transformation Problem Set 5B-7 5B-8 Plane Elasticity Formulas in Terms of Curvilinear Coordinates 5B-9 Complex Variable Solution for Plane Region Bounded by Circle in the z Plane Problem Set 5B References Bibliography CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES 6-1 6-2 6-3 6-4 Equilibrium Equations in Polar Coordinates... publisher including Bob Argentieri, Dan Magers, and the production team for their interest, cooperation, and help in publishing this book in a timely fashion, to James Chen for the checking and proofreading of the manuscript, as well as to Mike Plesniak of George Washington University and Jon Martin of NIST for providing an environment and culture conductive for scholarly pursuit CHAPTER 1 INTRODUCTORY... situation in continuum theories Contrary to continuum theories, temperature in MD is not an independent variable Instead, it is derivable from the velocities of atoms The treatment of temperature in molecular dynamics is incorporated in Chapter 4 Also the constitutive equations for soft biological tissues are included The readers can see that not only soft biological tissue can undergo large strains but... Energy-related solid mechanics: (i) High-temperature materials and coatings (ii) Fuel cells 1-1 TRENDS AND SCOPES 3 10 Advanced material processing: (i) High-speed machining (ii) Electronic and nanodevices, biodevices, biomaterials 11 Education in mechanics: (i) Need for multidisciplinary education between solid mechanics, physics, chemistry, and biology (ii) New mathematical skills in statistical mechanics and . ELASTICITY IN ENGINEERING MECHANICS Elasticity in Engineering Mechanics, Third Edition Arthur P. Boresi, Ken P. Chong and James D. Lee Copyright © 2011 John Wiley & Sons, Inc. ELASTICITY. Strain Tensor 94 2-10 Reciprocal Ellipsoid. Principal Strains. Strain Invariants 96 2-11 Determination of Principal Strains. Principal Axes 100 Problem Set 2-11 106 2-12 Determination of Strain Invariants. Volumetric. Sons, Inc. ELASTICITY IN ENGINEERING MECHANICS Third Edition ARTHUR P. BORESI Professor Emeritus University of Illinois, Urbana, Illinois and University of Wyoming, Laramie, Wyoming KEN P. CHONG Associate National