Introduction to Continuum Mechanics (eBook)

Front Cover 1
Introduction to Continuum Mechanics 4
Copyright Page 5
Table of Contents 6
Preface to the Fourth Edition 14
Chapter 1: Introduction 16
1.1 Introduction 16
1.2 What is Continuum Mechanics? 16
Chapter 2: Tensors 18
Part A: Indicial Notation 18
2.1 Summation Convention, Dummy Indices 18
2.2 Free Indices 19
2.3 The Kronecker Delta 20
2.4 The Permutation Symbol 21
2.5 Indicial Notation Manipulations 22
Problems For Part A 23
Part B: Tensors 24
2.6 Tensor: A Linear Transformation 24
2.7 Components of a Tensor 26
2.8 Components of a Transformed Vector 29
2.9 Sum of Tensors 31
2.10 Product of Two Tensors 31
2.11 Transpose of A Tensor 33
2.12 Dyadic Product of Vectors 34
2.13 Trace of A Tensor 35
2.14 Identity Tensor and Tensor Inverse 35
2.15 Orthogonal Tensors 37
2.16 Transformation Matrix Between Two Rectangular Cartesian Coordinate Systems 39
2.17 Transformation Law for Cartesian Components of A Vector 41
2.18 Transformation Law for Cartesian Components of a Tensor 42
2.19 Defining Tensor by Transformation Laws 44
2.20 Symmetric and Antisymmetric Tensors 46
2.21 The Dual Vector of an Antisymmetric Tensor 47
2.22 Eigenvalues and Eigenvectors of a Tensor 49
2.23 Principal Values and Principal Directions of Real Symmetric Tensors 53
2.24 Matrix of a Tensor with Respect to Principal Directions 54
2.25 Principal Scalar Invariants of a Tensor 55
Problems for Part B 56
Part C: Tensor Calculus 60
2.26 Tensor-Valued Functions of a Scalar 60
2.27 Scalar Field and Gradient of a Scalar Function 62
2.28 Vector Field and Gradient of a Vector Function 65
2.29 Divergence of a Vector Field and Divergence of a Tensor Field 65
2.30 Curl of a Vector Field 67
2.31 Laplacian of a Scalar Field 67
2.32 Laplacian of a Vector Field 67
Problems for Part C 68
Part D: Curvilinear Coordinates 69
2.33 Polar Coordinates 69
2.34 Cylindrical Coordinates 74
2.35 Spherical Coordinates 76
Problems for Part D 82
Chapter 3: Kinematics of a Continuum 84
3.1 Description of Motions of a Continuum 84
3.2 Material Description and Spatial Description 87
3.3 Material Derivative 89
3.4 Acceleration of a Particle 91
3.5 Displacement Field 96
3.6 Kinematic Equation for Rigid Body Motion 97
3.7 Infinitesimal Deformation 99
3.8 Geometrical Meaning of the Components of the Infinitesimal Strain Tensor 103
3.9 Principal Strain 108
3.10 Dilatation 108
3.11 The Infinitesimal Rotation Tensor 109
3.12 Time Rate of Change of a Material Element 110
3.13 The Rate of Deformation Tensor 110
3.14 The Spin Tensor and the Angular Velocity Vector 113
3.15 Equation of Conservation of Mass 114
3.16 Compatibility Conditions for Infinitesimal Strain Components 116
3.17 Compatibility Condition for Rate of Deformation Components 119
3.18 Deformation Gradient 120
3.19 Local Rigid Body Motion 121
3.20 Finite Deformation 121
3.21 Polar Decomposition Theorem 125
3.22 Calculation of Stretch and Rotation Tensors from the Deformation Gradient 126
3.23 Right Cauchy-Green Deformation Tensor 129
3.24 Lagrangian Strain Tensor 133
3.25 Left Cauchy-Green Deformation Tensor 136
3.26 Eulerian Strain Tensor 140
3.27 Change of Area Due to Deformation 143
3.28 Change of Volume Due to Deformation 144
3.29 Components of Deformation Tensors in Other Coordinates 146
3.30 Current Configuration as the Reference Configuration 154
Appendix 3.1: Necessary and Sufficient Conditions for Strain Compatibility 155
Appendix 3.2: Positive Definite Symmetric Tensors 158
Appendix 3.3: The Positive Definite Root of U2 = D 158
Problems for Chapter 3 160
Chapter 4: Stress and Integral Formulations of General Principles 170
4.1 Stress Vector 170
4.2 Stress Tensor 171
4.3 Components of Stress Tensor 173
4.4 Symmetry of Stress Tensor: Principle of Moment of Momentum 174
4.5 Principal Stresses 177
4.6 Maximum Shearing Stresses 177
4.7 Equations of Motion: Principle of Linear Momentum 183
4.8 Equations of Motion in Cylindrical and Spherical Coordinates 185
4.9 Boundary Condition for the Stress Tensor 186
4.10 Piola Kirchhoff Stress Tensors 189
4.11 Equations of Motion Written with Respect to the Reference Configuration 194
4.12 Stress Power 195
4.13 Stress Power in Terms of the Piola-Kirchhoff Stress Tensors 196
4.14 Rate of Heat Flow into a Differential Element by Conduction 198
4.15 Energy Equation 199
4.16 Entropy Inequality 200
4.17 Entropy Inequality in Terms of the Helmholtz Energy Function 201
4.18 Integral Formulations of the General Principles of Mechanics 202
Appendix 4.1: Determination of Maximum Shearing Stress and the Planes on Which It Acts 206
Problems for Chapter 4 209
Chapter 5: The Elastic Solid 216
5.1 Mechanical Properties 216
5.2 Linearly Elastic Solid 219
Part A: Isotropic Linearly Elastic Solid 222
5.3 Isotropic Linearly Elastic Solid 222
5.4 Young's Modulus, Poisson's Ratio, Shear Modulus, and Bulk Modulus 224
5.5 Equations of the Infinitesimal Theory of Elasticity 228
5.6 Navier Equations of Motion for Elastic Medium 230
5.7 Navier Equations in Cylindrical and Spherical Coordinates 231
5.8 Principle of Superposition 233
A.1 Plane Elastic Waves 233
5.9 Plane Irrotational Waves 233
5.10 Plane Equivoluminal Waves 236
5.11 Reflection of Plane Elastic Waves 241
5.12 Vibration of an Infinite Plate 244
A.2 Simple Extension, Torsion, and Pure Bending 246
5.13 Simple Extension 246
5.14 Torsion of a Circular Cylinder 249
5.15 Torsion of a Noncircular Cylinder: St. Venant's Problem 254
5.16 Torsion of Elliptical Bar 255
5.17 Prandtl's Formulation of the Torsion Problem 257
5.18 Torsion of a Rectangular Bar 261
5.19 Pure Bending of a Beam 262
A.3 Plane Stress and Plane Strain Solutions 265
5.20 Plane Strain Solutions 265
5.21 Rectangular Beam Bent by End Couples 268
5.22 Plane Stress Problem 269
5.23 Cantilever Beam with End Load 270
5.24 Simply Supported Beam Under Uniform Load 273
5.25 Slender Bar Under Concentrated Forces and St. Venant's Principle 275
5.26 Conversion for Strains Between Plane Strain and Plane Stress Solutions 277
5.27 Two-Dimensional Problems in Polar Coordinates 279
5.28 Stress Distribution Symmetrical About an Axis 280
5.29 Displacements for Symmetrical Stress Distribution in Plane Stress Solution 280
5.30 Thick-Walled Circular Cylinder Under Internal and External Pressure 282
5.31 Pure Bending of a Curved Beam 283
5.32 Initial Stress in a Welded Ring 285
5.33 Airy Stress Function phivf(r)cosntheta and phivf(r)sinntheta 285
5.34 Stress Concentration Due to a Small Circular Hole in a Plate Under Tension 289
5.35 Stress Concentration Due to a Small Circular Hole in a Plate Under Pure Shear 291
5.36 Simple Radial Distribution of Stresses in a Wedge Loaded at the Apex 292
5.37 Concentrated Line Load on a 2-D Half-Space: the Flamont Problem 293
A.4 Elastostatic Problems Solved with Potential Functions 294
5.38 Fundamental Potential Functions for Elastostatic Problems 294
5.39 Kelvin Problem: Concentrated Force at the Interior of an Infinite Elastic Space 305
5.40 Boussinesq Problem: Normal Concentrated Load on an Elastic Half-Space 308
5.41 Distributive Normal Load On The Surface Of An Elastic Half-Space 311
5.42 Hollow Sphere Subjected to Uniform Internal and External Pressure 312
5.43 Spherical Hole in a Tensile Field 313
5.44 Indentation by a Rigid Flat-Ended Smooth Indenter on an Elastic Half-Space 315
5.45 Indentation by a Smooth Rigid Sphere on an Elastic Half-Space 317
Appendix 5A.1: Solution of the Integral Equation in Section 5.45 321
Problems for Chapter 5, Part A, Sections 5.1-5.8 324
Problems for Chapter 5, Part A, Sections 5.9-5.12 (A.1) 325
Problems for Chapter 5, Part A, Sections 5.13-5.19 (A.2) 327
Problems for Chapter 5, Part A, Sections 5.20-5.37 (A.3) 330
Problems for Chapter 5, Part A, Sections 5.38-5.46 (A.4) 331
Part B: Anisotropic Linearly Elastic Solid 334
5.46 Constitutive Equations for an Anisotropic Linearly Elastic Solid 334
5.47 Plane of Material Symmetry 336
5.48 Constitutive Equation for a Monoclinic Linearly Elastic Solid 338
5.49 Constitutive Equation for an Orthotropic Linearly Elastic Solid 339
5.50 Constitutive Equation for a Transversely Isotropic Linearly Elastic Material 340
5.51 Constitutive Equation for an Isotropic Linearly Elastic Solid 342
5.52 Engineering Constants for an Isotropic Linearly Elastic Solid 343
5.53 Engineering Constants for a Transversely Isotropic Linearly Elastic Solid 344
5.54 Engineering Constants for an Orthotropic Linearly Elastic Solid 345
5.55 Engineering Constants for a Monoclinic Linearly Elastic Solid 347
Problems for Part B 348
Part C: Isotropic Elastic Solid Under Large Deformation 349
5.56 Change of Frame 349
5.57 Constitutive Equation for an Elastic Medium Under Large Deformation 353
5.58 Constitutive Equation for an Isotropic Elastic Medium 355
5.59 Simple Extension of an Incompressible Isotropic Elastic Solid 357
5.60 Simple Shear of an Incompressible Isotropic Elastic Rectangular Block 358
5.61 Bending of an Incompressible Isotropic Rectangular Bar 359
5.62 Torsion and Tension of an Incompressible Isotropic Solid Cylinder 362
Appendix 5C.1: Representation of Isotropic Tensor-Valued Functions 364
Problems for Part C 366
Chapter 6: Newtonian Viscous Fluid 368
6.1 Fluids 368
6.2 Compressible and Incompressible Fluids 369
6.3 Equations of Hydrostatics 370
6.4 Newtonian Fluids 372
6.5 Interpretation of lambda and mu 373
6.6 Incompressible Newtonian Fluid 374
6.7 Navier-Stokes Equations for Incompressible Fluids 375
6.8 Navier-Stokes Equations for Incompressible Fluids in Cylindrical and Spherical Coordinates 379
6.9 Boundary Conditions 380
6.10 Streamline, Pathline, Steady, Unsteady, Laminar, and Turbulent Flow 380
6.11 Plane Couette Flow 383
6.12 Plane Poiseuille Flow 383
6.13 Hagen-Poiseuille Flow 386
6.14 Plane Couette Flow of Two Layers of Incompressible Viscous Fluids 387
6.15 Couette Flow 389
6.16 Flow Near an Oscillating Plane 390
6.17 Dissipation Functions for Newtonian Fluids 391
6.18 Energy Equation for a Newtonian Fluid 393
6.19 Vorticity Vector 394
6.20 Irrotational Flow 396
6.21 Irrotational Flow of an Inviscid Incompressible Fluid of Homogeneous Density 397
6.22 Irrotational Flows as Solutions of Navier-Stokes Equation 399
6.23 Vorticity Transport Equation for Incompressible Viscous Fluid with a Constant Density 400
6.24 Concept of a Boundary Layer 403
6.25 Compressible Newtonian Fluid 404
6.26 Energy Equation in Terms of Enthalpy 405
6.27 Acoustic Wave 407
6.28 Irrotational, Barotropic Flows of an Inviscid Compressible Fluid 410
6.29 One-Dimensional Flow of a Compressible Fluid 413
6.30 Steady Flow of a Compressible Fluid Exiting a Large Tank Through a Nozzle 414
6.31 Steady Laminar Flow of a Newtonian Fluid in a Thin Elastic Tube: An Application to Pressure-Flow Relation in a Pulmonar... 416
Problems for Chapter 6 418
Chapter 7: The Reynolds Transport Theorem and Applications 426
7.1 Green's Theorem 426
7.2 Divergence Theorem 429
7.3 Integrals Over a Control Volume and Integrals Over a Material Volume 432
7.4 The Reynolds Transport Theorem 433
7.5 The Principle of Conservation of Mass 435
7.6 The Principle of Linear Momentum 437
7.7 Moving Frames 442
7.8 A Control Volume Fixed with Respect to a Moving Frame 445
7.9 The Principle of Moment of Momentum 445
7.10 The Principle of Conservation of Energy 447
7.11 The Entropy Inequality: The Second Law of Thermodynamics 451
Problems for Chapter 7 453
Chapter 8: Non-Newtonian Fluids 458
Part A: Linear Viscoelastic Fluid 459
8.1 Linear Maxwell Fluid 459
8.2 A Generalized Linear Maxwell Fluid with Discrete Relaxation Spectra 465
8.3 Integral Form of the Linear Maxwell Fluid and of the Generalized Linear Maxwell Fluid with Discrete Relaxation Spectra 466
8.4 A Generalized Linear Maxwell Fluid with a Continuous Relaxation Spectrum 467
8.5 Computation of Relaxation Spectrum and Relaxation Function 469
Part B: Nonlinear Viscoelastic Fluid 471
8.6 Current Configuration as Reference Configuration 471
8.7 Relative Deformation Gradient 472
8.8 Relative Deformation Tensors 474
8.9 Calculations of the Relative Deformation Tensor 475
8.10 History of the Relative Deformation Tensor and Rivlin-Ericksen Tensors 478
8.11 Rivlin-Ericksen Tensors in Terms of Velocity Gradient: The Recursive Formula 483
8.12 Relation between Velocity Gradient and Deformation Gradient 486
8.13 Transformation Law for the Relative Deformation Tensors Under a Change of Frame 486
8.14 Transformation Law for Rivlin-Ericksen Tensors Under a Change of Frame 489
8.15 Incompressible Simple Fluid 489
8.16 Special Single Integral-Type Nonlinear Constitutive Equations 490
8.17 General Single Integral-Type Nonlinear Constitutive Equations 493
8.18 Differential-Type Constitutive Equations for Incompressible Fluids 496
8.19 Objective Rate of Stress 498
8.20 Rate-Type Constitutive Equations 502
Part C: Viscometric Flow of an Incompressible simple Fluid 506
8.21 Viscometric Flow 506
8.22 Stresses in Viscometric Flow of an Incompressible Simple Fluid 508
8.23 Channel Flow 510
8.24 Couette Flow 512
Appendix 8.1: Gradient of Second-Order Tensor for Orthogonal Coordinates 516
Problems for Chapter 8 521
References 526
Answers to Problems 536
Chapter 2 536
Chapter 3 538
Chapter 4 542
Chapter 5 544
Chapter 6 546
Chapter 7 547
Chapter 8 548
Index 
528