كتاب Elasticity in Engineering Mechanics - Third Edition

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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

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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 Symbolsδij andijk (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 ofii.Strain
Definitions 86
2-8 Final Direction of Line Element.
Definition of Shearing Strain. Physical
Significance ofij(i=j) 89
Problem Set 2-8 94
2-9 Tensor Character ofαβ. 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 Solutionsof 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
Functionsψ(z) andχ(z) 428
5B-2 Displacement Components in Terms of
Analytic Functionsψ(z) andχ(z) 429
5B-3 Stress Components in Terms ofψ(z) and
χ(z) 430
5B-4 Expressions for Resultant Force and
Resultant Moment 433
5B-5 Mathematical Form of Functionsψ(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 FunctionF=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,∇
2
F=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 VectorF 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 VectorkZand 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

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