ELASTICITY IN ENGINEERINGMECHANICS

Elasticity in Engineering Mechanics, Third Edition Arthur P. Boresi, Ken P. Chong and James D. LeeCopyright © 2011 John Wiley & Sons, Inc.

ELASTICITY INENGINEERINGMECHANICS

Third Edition

ARTHUR P. BORESIProfessor EmeritusUniversity of Illinois, Urbana, Illinois

and

University of Wyoming, Laramie, Wyoming

KEN P. CHONGAssociateNational Institute of Standards and Technology, Gaithersburg, Maryland

and

ProfessorDepartment of Mechanical and Aerospace EngineeringGeorge Washington University, Washington, D.C.

JAMES D. LEEProfessorDepartment of Mechanical and Aerospace EngineeringGeorge 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 2011620.1′1232–dc22

2010030995

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

CONTENTS

Preface xvii

CHAPTER 1 INTRODUCTORY CONCEPTS AND MATHEMATICS 1

Part I Introduction 1

1-1 Trends and Scopes 11-2 Theory of Elasticity 71-3 Numerical Stress Analysis 81-4 General Solution of the Elasticity

Problem 91-5 Experimental Stress Analysis 91-6 Boundary Value Problems of Elasticity 10

Part II Preliminary Concepts 11

1-7 Brief Summary of Vector Algebra 121-8 Scalar Point Functions 161-9 Vector Fields 181-10 Differentiation of Vectors 191-11 Differentiation of a Scalar Field 211-12 Differentiation of a Vector Field 211-13 Curl of a Vector Field 221-14 Eulerian Continuity Equation for Fluids 22

v

vi CONTENTS

1-15 Divergence Theorem 251-16 Divergence Theorem in Two

Dimensions 271-17 Line and Surface Integrals (Application of

Scalar Product) 281-18 Stokes’s Theorem 291-19 Exact Differential 301-20 Orthogonal Curvilinear Coordiantes in

Three-Dimensional Space 311-21 Expression for Differential Length in

Orthogonal Curvilinear Coordinates 321-22 Gradient and Laplacian in Orthogonal

Curvilinear Coordinates 33

Part III Elements of Tensor Algebra 36

1-23 Index Notation: Summation Convention 361-24 Transformation of Tensors under Rotation

of Rectangular Cartesian CoordinateSystem 40

1-25 Symmetric and Antisymmetric Parts of aTensor 46

1-26 Symbols δij and εijk (the Kronecker Deltaand the Alternating Tensor) 47

1-27 Homogeneous Quadratic Forms 491-28 Elementary Matrix Algebra 521-29 Some Topics in the Calculus of

Variations 56References 60Bibliography 63

CHAPTER 2 THEORY OF DEFORMATION 65

2-1 Deformable, Continuous Media 652-2 Rigid-Body Displacements 662-3 Deformation of a Continuous Region.

Material Variables. Spatial Variables 682-4 Restrictions on Continuous Deformation

of a Deformable Medium 71Problem Set 2-4 75

2-5 Gradient of the Displacement Vector.Tensor Quantity 76

CONTENTS vii

2-6 Extension of an Infinitesimal Line Element 78Problem Set 2-6 85

2-7 Physical Significance of εii . StrainDefinitions 86

2-8 Final Direction of Line Element.Definition of Shearing Strain. PhysicalSignificance of εij(i �= j) 89Problem Set 2-8 94

2-9 Tensor Character of εαβ . Strain Tensor 942-10 Reciprocal Ellipsoid. Principal Strains.

Strain Invariants 962-11 Determination of Principal Strains.

Principal Axes 100Problem Set 2-11 106

2-12 Determination of Strain Invariants.Volumetric Strain 108

2-13 Rotation of a Volume Element. Relation toDisplacement Gradients 113Problem Set 2-13 116

2-14 Homogeneous Deformation 1182-15 Theory of Small Strains and Small Angles

of Rotation 121Problem Set 2-15 130

2-16 Compatibility Conditions of the ClassicalTheory of Small Displacements 132Problem Set 2-16 137

2-17 Additional Conditions Imposed byContinuity 138

2-18 Kinematics of Deformable Media 140Problem Set 2-18 146

Appendix 2A Strain–Displacement Relations in OrthogonalCurvilinear Coordinates 146

2A-1 Geometrical Preliminaries 1462A-2 Strain–Displacement Relations 148

Appendix 2B Derivation of Strain–Displacement Relations forSpecial Coordinates by Cartesian Methods 151

2B-1 Cylindrical Coordinates 1512B-2 Oblique Straight-Line Coordinates 153

viii CONTENTS

Appendix 2C Strain–Displacement Relations in GeneralCoordinates 155

2C-1 Euclidean Metric Tensor 1552C-2 Strain Tensors 157References 159Bibliography 160

CHAPTER 3 THEORY OF STRESS 161

3-1 Definition of Stress 1613-2 Stress Notation 1643-3 Summation of Moments. Stress at a Point.

Stress on an Oblique Plane 166Problem Set 3-3 171

3-4 Tensor Character of Stress. Transformationof Stress Components under Rotation ofCoordinate Axes 175Problem Set 3-4 179

3-5 Principal Stresses. Stress Invariants.Extreme Values 179Problem Set 3-5 183

3-6 Mean and Deviator Stress Tensors.Octahedral Stress 184Problem Set 3-6 189

3-7 Approximations of Plane Stress. Mohr’sCircles in Two and Three Dimensions 193Problem Set 3-7 200

3-8 Differential Equations of Motion of aDeformable Body Relative to SpatialCoordinates 201Problem Set 3-8 205

Appendix 3A Differential Equations of Equilibrium in CurvilinearSpatial Coordinates 207

3A-1 Differential Equations of Equilibrium inOrthogonal Curvilinear SpatialCoordinates 207

3A-2 Specialization of Equations of Equilibrium 2083A-3 Differential Equations of Equilibrium in

General Spatial Coordinates 210

CONTENTS ix

Appendix 3B Equations of Equilibrium Including Couple Stressand Body Couple 211

Appendix 3C Reduction of Differential Equations of Motion forSmall-Displacement Theory 214

3C-1 Material Derivative. Material Derivativeof a Volume Integral 214

3C-2 Differential Equations of EquilibriumRelative to Material Coordinates 218

References 224Bibliography 225

CHAPTER 4 THREE-DIMENSIONAL EQUATIONS OFELASTICITY 226

4-1 Elastic and Nonelastic Response of a Solid 2264-2 Intrinsic Energy Density Function

(Adiabatic Process) 2304-3 Relation of Stress Components to Strain

Energy Density Function 232Problem Set 4-3 240

4-4 Generalized Hooke’s Law 241Problem Set 4-4 255

4-5 Isotropic Media. Homogeneous Media 2554-6 Strain Energy Density for Elastic Isotropic

Medium 256Problem Set 4-6 262

4-7 Special States of Stress 266Problem Set 4-7 268

4-8 Equations of Thermoelasticity 2694-9 Differential Equation of Heat Conduction 2704-10 Elementary Approach to Thermal-Stress

Problem in One and Two Variables 272Problem 276

4-11 Stress–Strain–Temperature Relations 276Problem Set 4-11 283

4-12 Thermoelastic Equations in Terms ofDisplacement 285

4-13 Spherically Symmetrical StressDistribution (The Sphere) 294Problem Set 4-13 299

x CONTENTS

4-14 Thermoelastic Compatibility Equations inTerms of Components of Stress andTemperature. Beltrami–MichellRelations 299Problem Set 4-14 304

4-15 Boundary Conditions 305Problem Set 4-15 310

4-16 Uniqueness Theorem for EquilibriumProblem of Elasticity 311

4-17 Equations of Elasticity in Terms ofDisplacement Components 314Problem Set 4-17 316

4-18 Elementary Three-Dimensional Problemsof Elasticity. Semi-Inverse Method 317Problem Set 4-18 323

4-19 Torsion of Shaft with Constant CircularCross Section 327Problem Set 4-19 331

4-20 Energy Principles in Elasticity 3324-21 Principle of Virtual Work 333

Problem Set 4-21 3384-22 Principle of Virtual Stress (Castigliano’s

Theorem) 3394-23 Mixed Virtual Stress–Virtual Strain

Principles (Reissner’s Theorem) 342

Appendix 4A Application of the Principle of Virtual Work to aDeformable Medium (Navier–Stokes Equations) 343

Appendix 4B Nonlinear Constitutive Relationships 345

4B-1 Variable Stress–Strain Coefficients 3464B-2 Higher-Order Relations 3464B-3 Hypoelastic Formulations 3464B-4 Summary 347

Appendix 4C Micromorphic Theory 347

4C-1 Introduction 3474C-2 Balance Laws of Micromorphic Theory 3504C-3 Constitutive Equations of Micromorphic

Elastic Solid 351

CONTENTS xi

Appendix 4D Atomistic Field Theory 352

4D-1 Introduction 3534D-2 Phase-Space and Physical-Space

Descriptions 3534D-3 Definitions of Atomistic Quantities in

Physical Space 3554D-4 Conservation Equations 357References 359Bibliography 364

CHAPTER 5 PLANE THEORY OF ELASTICITY INRECTANGULAR CARTESIAN COORDINATES 365

5-1 Plane Strain 365Problem Set 5-1 370

5-2 Generalized Plane Stress 371Problem Set 5-2 376

5-3 Compatibility Equation in Terms of StressComponents 377Problem Set 5-3 382

5-4 Airy Stress Function 383Problem Set 5-4 392

5-5 Airy Stress Function in Terms ofHarmonic Functions 399

5-6 Displacement Components for PlaneElasticity 401Problem Set 5-6 404

5-7 Polynomial Solutions of Two-DimensionalProblems in Rectangular CartesianCoordinates 408Problem Set 5-7 411

5-8 Plane Elasticity in Terms of DisplacementComponents 415Problem Set 5-8 416

5-9 Plane Elasticity Relative to ObliqueCoordinate Axes 416

Appendix 5A Plane Elasticity with Couple Stresses 420

5A-1 Introduction 4205A-2 Equations of Equilibrium 421

xii CONTENTS

5A-3 Deformation in Couple Stress Theory 4215A-4 Equations of Compatibility 4255A-5 Stress Functions for Plane Problems with

Couple Stresses 426

Appendix 5B Plane Theory of Elasticity in Terms of ComplexVariables 428

5B-1 Airy Stress Function in Terms of AnalyticFunctions ψ(z) and χ(z) 428

5B-2 Displacement Components in Terms ofAnalytic Functions ψ(z) and χ(z) 429

5B-3 Stress Components in Terms of ψ(z) andχ(z) 430

5B-4 Expressions for Resultant Force andResultant Moment 433

5B-5 Mathematical Form of Functions ψ(z) andχ(z) 434

5B-6 Plane Elasticity Boundary Value Problemsin Complex Form 438

5B-7 Note on Conformal Transformation 440Problem Set 5B-7 445

5B-8 Plane Elasticity Formulas in Terms ofCurvilinear Coordinates 445

5B-9 Complex Variable Solution for PlaneRegion Bounded by Circle in thez Plane 448Problem Set 5B 452

References 453Bibliography 454

CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES 455

6-1 Equilibrium Equations in PolarCoordinates 455

6-2 Stress Components in Terms of AiryStress Function F = F(r, θ) 456

6-3 Strain–Displacement Relations in PolarCoordinates 457Problem Set 6-3 460

6-4 Stress–Strain–Temperature Relations 461Problem Set 6-4 462

CONTENTS xiii

6-5 Compatibility Equation for PlaneElasticity in Terms of Polar Coordinates 463Problem Set 6-5 464

6-6 Axially Symmetric Problems 467Problem Set 6-6 483

6-7 Plane Elasticity Equations in Terms ofDisplacement Components 485

6-8 Plane Theory of Thermoelasticity 489Problem Set 6-8 492

6-9 Disk of Variable Thickness andNonhomogeneous Anisotropic Material 494Problem Set 6-9 497

6-10 Stress Concentration Problem of CircularHole in Plate 498Problem Set 6-10 504

6-11 Examples 505Problem Set 6-11 510

Appendix 6A Stress–Couple Theory of Stress ConcentrationResulting from Circular Hole in Plate 519

Appendix 6B Stress Distribution of a Diametrically CompressedPlane Disk 522

References 525

CHAPTER 7 PRISMATIC BAR SUBJECTED TO END LOAD 527

7-1 General Problem of Three-DimensionalElastic Bars Subjected to Transverse EndLoads 527

7-2 Torsion of Prismatic Bars. Saint-Venant’sSolution. Warping Function 529Problem Set 7-2 534

7-3 Prandtl Torsion Function 534Problem Set 7-3 538

7-4 A Method of Solution of the TorsionProblem: Elliptic Cross Section 538Problem Set 7-4 542

7-5 Remarks on Solutions of the LaplaceEquation, ∇2F = 0 542Problem Set 7-5 544

xiv CONTENTS

7-6 Torsion of Bars with Tubular Cavities 547Problem Set 7-6 549

7-7 Transfer of Axis of Twist 5497-8 Shearing–Stress Component in Any

Direction 550Problem Set 7-8 554

7-9 Solution of Torsion Problem by thePrandtl Membrane Analogy 554Problem Set 7-9 561

7-10 Solution by Method of Series. RectangularSection 562Problem Set 7-10 566

7-11 Bending of a Bar Subjected to TransverseEnd Force 569Problem Set 7-11 577

7-12 Displacement of a Cantilever BeamSubjected to Transverse End Force 577Problem Set 7-12 581

7-13 Center of Shear 581Problem Set 7-13 582

7-14 Bending of a Bar with Elliptic CrossSection 584

7-15 Bending of a Bar with Rectangular CrossSection 586Problem Set 7-15 590Review Problems 590

Appendix 7A Analysis of Tapered Beams 591

References 595

CHAPTER 8 GENERAL SOLUTIONS OF ELASTICITY 597

8-1 Introduction 597Problem Set 8-1 598

8-2 Equilibrium Equations 598Problem Set 8-2 600

8-3 The Helmholtz Transformation 600Problem Set 8-3 601

8-4 The Galerkin (Papkovich) Vector 602Problem Set 8-4 603

CONTENTS xv

8-5 Stress in Terms of the Galerkin Vector F 603Problem Set 8-5 604

8-6 The Galerkin Vector: A Solution of theEquilibrium Equations of Elasticity 604Problem Set 8-6 606

8-7 The Galerkin Vector kZ and Love’s StrainFunction for Solids of Revolution 606Problem Set 8-7 608

8-8 Kelvin’s Problem: Single Force Applied inthe Interior of an Infinitely Extended Solid 609Problem Set 8-8 610

8-9 The Twinned Gradient and Its Applicationto Determine the Effects of a Change ofPoisson’s Ratio 611

8-10 Solutions of the Boussinesq and CerrutiProblems by the Twinned GradientMethod 614Problem Set 8-10 617

8-11 Additional Remarks onThree-Dimensional Stress Functions 617

References 618Bibliography 619

INDEX 621

PREFACE

The material presented is intended to serve as a basis for a critical study of the fun-damentals of elasticity and several branches of solid mechanics, including advancedmechanics of materials, theories of plates and shells, composite materials, plasticitytheory, finite element, and other numerical methods as well as nanomechanics andbiomechanics. In the 21st century, the transcendent and translational technologiesinclude nanotechnology, microelectronics, information technology, and biotechnol-ogy as well as the enabling and supporting mechanical and civil infrastructuresystems and smart materials. These technologies are the primary drivers of thecentury and the new economy in a modern society.

Chapter 1 includes, for ready reference, new trends, research needs, and certainmathematic preliminaries. Depending on the background of the reader, this materialmay be used either as required reading or as reference material. The main content ofthe book begins with the theory of deformation in Chapter 2. Although the majorityof the book is focused on stress–strain theory, the concept of deformation with largestrains (Cauchy strain tensor and Green–Saint-Venant strain tensor) is included. Thetheory of stress is presented in Chapter 3. The relations among different stress mea-sures, namely, Cauchy stress tensor, first- and second-order Piola–Kirchhoff stresstensors, are described. Molecular dynamics (MD) views a material body as a col-lection of a huge but finite number of different kinds of atoms. It is emphasized thatMD is the heart of nanoscience and technology, and it deals with material propertiesand behavior at the atomistic scale. The differential equations of motion of MD areintroduced. The readers may see the similarity and the difference between a contin-uum theory and an atomistic theory clearly. The theories of deformation and stressare treated separately to emphasize their independence of one another and alsoto emphasize their mathematical similarity. By so doing, one can clearly see that

xvii

xviii PREFACE

these theories depend only on approximations related to modeling of a continuousmedium, and that they are independent of material behavior. The theories of defor-mation and stress are united in Chapter 4 by the introduction of three-dimensionalstress–strain–temperature relations (constitutive relations). The constitutive rela-tions in MD, through interatomic potentials, are introduced. The force–positionrelation between atoms is nonlinear and nonlocal, which is contrary to the situationin continuum theories. Contrary to continuum theories, temperature in MD is notan independent variable. Instead, it is derivable from the velocities of atoms. Thetreatment of temperature in molecular dynamics is incorporated in Chapter 4. Alsothe constitutive equations for soft biological tissues are included. The readers cansee that not only soft biological tissue can undergo large strains but also exert anactive stress, which is the fundamental difference between lifeless material and liv-ing biological tissue. The significance of active stress is demonstrated through anexample in Chapter 6. The major portion of Chapter 4 is devoted to linearly elasticmaterials. However, discussions of nonlinear constitutive relations, micromorphictheory, and concurrent atomistic/continuum theory are presented in Appendices4B, 4C, and 4D, respectively. Chapters 5 and 6 treat the plane theory of elasticity,in rectangular and polar coordinates, respectively. Chapter 7 presents the three-dimensional problem of prismatic bars subjected to end loads. Material on thermalstresses is incorporated in a logical manner in the topics of Chapters 4, 5, and 6.

General solutions of elasticity are presented in Chapter 8. Extensive use is madeof appendixes for more advanced topics such as complex variables (Appendix 5B)and stress–couple theory (Appendixes 5A and 6A). In addition, in each chapter,examples and problems are given, along with explanatory notes, references, and abibliography for further study.

As presented, the book is valuable as a text for students and as a reference forpracticing engineers/scientists. The material presented here may be used for severaldifferent types of courses. For example, a semester course for senior engineeringstudents may include topics from Chapter 2 (Sections 2-1 through 2-16), Chapter 3(Sections 3-1 through 3-8), Chapter 4 (Sections 4-1 through 4-7 and Sections 4-9through 4-12), Chapter 5 (Sections 5-1 through 5-7), as much as possible fromChapter 6 (from Sections 6-1 through Section 6-6), and considerable problem solv-ing. A quarter course for seniors could cover similar material from Chapters 2through 5, with less emphasis on the examples and problem solving. A course forfirst-year graduate students in civil and mechanical engineering and related engi-neering fields can include Chapters 1 through 6, with selected materials from theappendixes and/or Chapters 7 and 8. A follow-up graduate course can include mostof 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 students.

Special thanks are due to the publisher including Bob Argentieri, Dan Magers,and the production team for their interest, cooperation, and help in publishing thisbook in a timely fashion, to James Chen for the checking and proofreading of themanuscript, as well as to Mike Plesniak of George Washington University and JonMartin of NIST for providing an environment and culture conductive for scholarlypursuit.