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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.
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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. 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
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