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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

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 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 δij 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

xii CONTENTS

5A-3 Deformation in Couple Stress Theory 421

5A-4 Equations of Compatibility 425

5A-5 Stress Functions for Plane Problems with

Couple Stresses 426

Appendix 5B Plane Theory of Elasticity in Terms of Complex

Variables 428

5B-1 Airy Stress Function in Terms of Analytic

Functions ψ(z) and χ(z) 428

5B-2 Displacement Components in Terms of

Analytic Functions ψ(z) and χ(z) 429

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

χ(z) 430

5B-4 Expressions for Resultant Force and

Resultant Moment 433

5B-5 Mathematical Form of Functions ψ(z) and

χ(z) 434

5B-6 Plane Elasticity Boundary Value Problems

in Complex Form 438

5B-7 Note on Conformal Transformation 440

Problem Set 5B-7 445

5B-8 Plane Elasticity Formulas in Terms of

Curvilinear Coordinates 445

5B-9 Complex Variable Solution for Plane

Region Bounded by Circle in the

z Plane 448

Problem Set 5B 452

References 453

Bibliography 454

CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES 455

6-1 Equilibrium Equations in Polar

Coordinates 455

6-2 Stress Components in Terms of Airy

Stress Function F = F (r, θ ) 456

6-3 Strain–Displacement Relations in Polar

Coordinates 457

Problem Set 6-3 460

6-4 Stress–Strain–Temperature Relations 461

Problem Set 6-4 462

CONTENTS xiii

6-5 Compatibility Equation for Plane

Elasticity in Terms of Polar Coordinates 463

Problem Set 6-5 464

6-6 Axially Symmetric Problems 467

Problem Set 6-6 483

6-7 Plane Elasticity Equations in Terms of

Displacement Components 485

6-8 Plane Theory of Thermoelasticity 489

Problem Set 6-8 492

6-9 Disk of Variable Thickness and

Nonhomogeneous Anisotropic Material 494

Problem Set 6-9 497

6-10 Stress Concentration Problem of Circular

Hole in Plate 498

Problem Set 6-10 504

6-11 Examples 505

Problem Set 6-11 510

Appendix 6A Stress–Couple Theory of Stress Concentration

Resulting from Circular Hole in Plate 519

Appendix 6B Stress Distribution of a Diametrically Compressed

Plane Disk 522

References 525

CHAPTER 7 PRISMATIC BAR SUBJECTED TO END LOAD 527

7-1 General Problem of Three-Dimensional

Elastic Bars Subjected to Transverse End

Loads 527

7-2 Torsion of Prismatic Bars. Saint-Venant’s

Solution. Warping Function 529

Problem Set 7-2 534

7-3 Prandtl Torsion Function 534

Problem Set 7-3 538

7-4 A Method of Solution of the Torsion

Problem: Elliptic Cross Section 538

Problem Set 7-4 542

7-5 Remarks on Solutions of the Laplace

Equation, ∇2F = 0 542

Problem Set 7-5 544

xiv CONTENTS

7-6 Torsion of Bars with Tubular Cavities 547

Problem Set 7-6 549

7-7 Transfer of Axis of Twist 549

7-8 Shearing–Stress Component in Any

Direction 550

Problem Set 7-8 554

7-9 Solution of Torsion Problem by the

Prandtl Membrane Analogy 554

Problem Set 7-9 561

7-10 Solution by Method of Series. Rectangular

Section 562

Problem Set 7-10 566

7-11 Bending of a Bar Subjected to Transverse

End Force 569

Problem Set 7-11 577

7-12 Displacement of a Cantilever Beam

Subjected to Transverse End Force 577

Problem Set 7-12 581

7-13 Center of Shear 581

Problem Set 7-13 582

7-14 Bending of a Bar with Elliptic Cross

Section 584

7-15 Bending of a Bar with Rectangular Cross

Section 586

Problem Set 7-15 590

Review Problems 590

Appendix 7A Analysis of Tapered Beams 591

References 595

CHAPTER 8 GENERAL SOLUTIONS OF ELASTICITY 597

8-1 Introduction 597

Problem Set 8-1 598

8-2 Equilibrium Equations 598

Problem Set 8-2 600

8-3 The Helmholtz Transformation 600

Problem Set 8-3 601

8-4 The Galerkin (Papkovich) Vector 602

Problem Set 8-4 603

CONTENTS xv

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

Problem Set 8-5 604

8-6 The Galerkin Vector: A Solution of the

Equilibrium Equations of Elasticity 604

Problem Set 8-6 606

8-7 The Galerkin Vector kZ and Love’s Strain

Function for Solids of Revolution 606

Problem Set 8-7 608

8-8 Kelvin’s Problem: Single Force Applied in

the Interior of an Infinitely Extended Solid 609

Problem Set 8-8 610

8-9 The Twinned Gradient and Its Application

to Determine the Effects of a Change of

Poisson’s Ratio 611

8-10 Solutions of the Boussinesq and Cerruti

Problems by the Twinned Gradient

Method 614

Problem Set 8-10 617

8-11 Additional Remarks on

Three-Dimensional Stress Functions 617

References 618

Bibliography 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 advanced

mechanics of materials, theories of plates and shells, composite materials, plasticity

theory, finite element, and other numerical methods as well as nanomechanics and

biomechanics. In the 21st century, the transcendent and translational technologies

include nanotechnology, microelectronics, information technology, and biotechnol￾ogy as well as the enabling and supporting mechanical and civil infrastructure

systems and smart materials. These technologies are the primary drivers of the

century and the new economy in a modern society.

Chapter 1 includes, for ready reference, new trends, research needs, and certain

mathematic preliminaries. Depending on the background of the reader, this material

may be used either as required reading or as reference material. The main content of

the book begins with the theory of deformation in Chapter 2. Although the majority

of the book is focused on stress–strain theory, the concept of deformation with large

strains (Cauchy strain tensor and Green–Saint-Venant strain tensor) is included. The

theory of stress is presented in Chapter 3. The relations among different stress mea￾sures, namely, Cauchy stress tensor, first- and second-order Piola–Kirchhoff stress

tensors, 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 that

MD is the heart of nanoscience and technology, and it deals with material properties

and behavior at the atomistic scale. The differential equations of motion of MD are

introduced. The readers may see the similarity and the difference between a contin￾uum theory and an atomistic theory clearly. The theories of deformation and stress

are treated separately to emphasize their independence of one another and also

to emphasize their mathematical similarity. By so doing, one can clearly see that

xvii