Continuum mechanics

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Title:

Continuum mechanics

Personal Author:

Irgens, Fridtjov, 1935-

Publication Information:

Berlin : Springer, 2008

Physical Description:

xviii, 661 p. : ill. ; 24 cm.

ISBN:

9783540742975

Subject Term:

Mechanics

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010177807	QA808.2 I73 2008	Open Access Book	Book	Searching... Unknown

This book presents an introduction into the entire science of Continuum Mechanics in three parts. The presentation is modern and comprehensive. Its introduction into tensors is very gentle. The book contains many examples and exercises, and is intended for scientists, practitioners and students of mechanics.

1 Introduction	p. 1
1.1 The Continuum Hypothesis	p. 1
1.2 Elasticity, Plasticity, and Fracture	p. 2
1.3 Fluids	p. 8
1.4 Viscoelasticity	p. 12
1.5 An Outline for the Present Book	p. 16
2 Mathematical Foundation	p. 19
2.1 Matrices and Determinants	p. 19
2.2 Coordinate Systems and Vectors	p. 23
2.3 Coordinate Transformations	p. 28
2.4 Scalar Fields and Vector Fields	p. 30
Problems	p. 34
3 Dynamics	p. 37
3.1 Kinematics	p. 37
3.1.1 Lagrangian Coordinates and Eulerian Coordinates	p. 37
3.1.2 Material Derivative of an Intensive Quantity	p. 40
3.1.3 Material Derivative of an Extensive Quantity	p. 41
3.2 Equations of Motion	p. 42
3.2.1 Euler's Axioms	p. 42
3.2.2 Newton's 3. Law	p. 46
3.2.3 Coordinate Stresses	p. 47
3.2.4 Cauchy's Stress Theorem and Cauchy's Stress Tensor	p. 50
3.2.5 Cauchy Equations of Motion	p. 54
3.2.6 Alternative Derivation of the Cauchy Equations	p. 58
3.3 Stress Analysis	p. 60
3.3.1 Principal Stresses	p. 60
3.3.2 Stress Deviator and Stress Isotrop	p. 65
3.3.3 Extremal Values for Normal Stress	p. 68
3.3.4 Maximum Shear Stress	p. 69
3.3.5 Plane Stress	p. 70
3.3.6 Mohr Diagram for Plane Stress	p. 73
3.3.7 Mohr Diagram for General States of Stress	p. 77
Problems	p. 79
4 Tensors	p. 83
4.1 Definition of Tensors	p. 83
4.2 Tensor Algebra	p. 89
4.2.1 Isotropic Tensors of 4. Order	p. 94
4.2.2 Tensors as Polyadics	p. 96
4.3 Tensors of 2. Order. Part One	p. 97
4.3.1 Symmetric Tensors of 2. Order	p. 100
4.3.2 Alternative Invariants	p. 103
4.3.3 Deviator and Isotrop	p. 104
4.4 Tensor Fields	p. 105
4.4.1 Gradient, Divergence, and Rotation	p. 105
4.4.2 Del-Operator	p. 107
4.4.3 Orthogonal Coordinates	p. 108
4.4.4 Material Derivative of a Tensor Field	p. 110
4.5 Rigid-Body Dynamics	p. 111
4.5.1 Kinematics	p. 112
4.5.2 Relative Motion	p. 117
4.5.3 Kinetics	p. 119
4.6 Tensors of 2. Order. Part Two	p. 122
4.6.1 Rotation of Vectors and Tensors	p. 123
4.6.2 Polar Decomposition	p. 124
4.6.3 Isotropic Functions of Tensors	p. 125
Problems	p. 129
5 Deformation Analysis	p. 133
5.1 Strain Measures	p. 133
5.2 The Green Strain Tensor	p. 134
5.3 Small Strains and Small Deformations	p. 139
5.3.1 Small Strains	p. 140
5.3.2 Small Deformations	p. 141
5.3.3 Coordinate Strains in Cylindrical Coordinates and Spherical Coordinates	p. 142
5.3.4 Principal Strains and Principal Directions of Strains	p. 144
5.3.5 Strain Isotrop and Strain Deviator	p. 145
5.3.6 Rotation Tensor for Small Deformations	p. 145
5.3.7 Small Strains in a Material Surface	p. 147
5.3.8 Mohr Diagram for Strain	p. 148
5.3.9 Equations of Compatibility	p. 148
5.3.10 Compatibility Equations as Sufficient Conditions	p. 150
5.4 Rates of Deformation, Strain, and Rotation	p. 151
5.4.1 Rate of Deformation Matrix and Rate of Rotation Matrix in Cylindrical and Spherical Coordinates	p. 158
5.5 Large Deformations	p. 160
5.5.1 Special Types of Deformations and Flows	p. 166
5.5.2 The Continuity Equation in a Particle	p. 172
5.5.3 Reduction to Small Deformations	p. 172
5.5.4 Deformation with Respect to the Present Configuration	p. 173
5.6 The Piola-Kirchhoff Stress Tensors	p. 175
Problems	p. 178
6 Work and Energy	p. 183
6.1 Mechanical Energy Balance	p. 183
6.1.1 The Work-Energy Equation for Rigid Bodies	p. 186
6.1.2 Conjugate Stress Tensors and Deformation Tensors	p. 189
6.2 The Principle of Virtual Power	p. 190
6.3 Thermal Energy Balance	p. 192
6.3.1 Thermodynamic Introduction	p. 192
6.3.2 Thermal Energy Balance	p. 193
6.4 The Second Law of Thermodynamics	p. 195
Problems	p. 198
7 Theory of Elasticity	p. 199
7.1 Introduction	p. 199
7.2 The Hookean Solid	p. 200
7.2.1 An Alternative Development of the Generalized Hooke's Law	p. 205
7.2.2 Strain Energy	p. 206
7.3 Two-Dimensional Theory of Elasticity	p. 207
7.3.1 Plane Stress	p. 207
7.3.2 Plane Displacements	p. 213
7.3.3 Airy's Stress Function	p. 217
7.3.4 Airy's Stress Function in Polar Coordinates	p. 223
7.3.5 Axial Symmetry	p. 229
7.4 Torsion of Cylindrical Bars	p. 232
7.4.1 The Coulomb Theory of Torsion	p. 232
7.4.2 The Saint-Venant Theory of Torsion	p. 234
7.4.3 Prandtl's Stress Function	p. 238
7.4.4 The Membrane Analogy	p. 241
7.5 Thermoelasticity	p. 244
7.5.1 Constitutive Equations	p. 244
7.5.2 Plane Stress	p. 245
7.5.3 Plane Displacements	p. 248
7.6 Hyperelasticity	p. 249
7.6.1 Elastic Energy	p. 249
7.6.2 The Basic Equations of Hyperelasticity	p. 252
7.6.3 The Uniqueness Theorem	p. 258
7.7 Stress Waves in Elastic Materials	p. 260
7.7.1 Longitudinal Waves in Cylindrical Bars	p. 260
7.7.2 The Hopkinson Experiment	p. 266
7.7.3 Plane Elastic Waves	p. 268
7.7.4 Elastic Waves in an Infinite Medium	p. 270
7.7.5 Seismic Waves	p. 270
7.7.6 Reflection of Elastic Waves	p. 271
7.7.7 Tensile Fracture Due to Compression Wave	p. 272
7.7.8 Surface Waves. Rayleigh Waves	p. 273
7.8 Anisotropic Materials	p. 274
7.8.1 Hyperelasticity	p. 276
7.8.2 Materials with one Plane of Symmetry	p. 277
7.8.3 Three Orthogonal Symmetry Planes. Orthotropy	p. 279
7.8.4 Transverse Isotropy	p. 281
7.8.5 Isotropy	p. 283
7.9 Composite Materials	p. 284
7.9.1 Lamina	p. 285
7.9.2 From Lamina Axes to Laminate Axes	p. 288
7.9.3 Engineering Parameters Related to Laminate Axes	p. 290
7.9.4 Plate Laminate of Unidirectional Laminas	p. 290
7.10 Large Deformations	p. 292
7.10.1 Isotropic Elasticity	p. 293
7.10.2 Hyperelasticity	p. 294
Problems	p. 297
8 Fluid Mechanics	p. 303
8.1 Introduction	p. 303
8.2 Control Volume. Reynolds' Transport Theorem	p. 306
8.2.1 Alternative Derivation of the Reynolds' Transport Theorem	p. 309
8.2.2 Control Volume Equations	p. 310
8.2.3 Continuity Equation	p. 312
8.3 Perfect Fluid [identical with] Eulerian Fluid	p. 313
8.3.1 Bernoulli's Equation	p. 315
8.3.2 Circulation and Vorticity	p. 319
8.3.3 Sound Waves	p. 322
8.4 Linearly Viscous Fluid = Newtonian Fluid	p. 323
8.4.1 Constitutive Equations	p. 323
8.4.2 The Navier-Stokes Equations	p. 330
8.4.3 Dissipation	p. 333
8.4.4 The Energy Equation	p. 335
8.4.5 The Bernoulli Equation for Pipe Flow	p. 336
8.5 Potential Flow	p. 339
8.5.1 The D'alembert Paradox	p. 343
8.6 Non-Newtonian Fluids	p. 343
8.6.1 Introduction	p. 343
8.6.2 Generalized Newtonian Fluids	p. 344
8.6.3 Viscometric Flows. Kinematics	p. 347
8.6.4 Material Functions for Viscometric Flows	p. 353
8.6.5 Extensional Flows	p. 356
Problems	p. 358
9 Viscoelasticity	p. 361
9.1 Introduction	p. 361
9.2 Linearly Viscoelastic Materials	p. 368
9.2.1 Mechanical Models	p. 368
9.2.2 General Response Equation	p. 376
9.2.3 The Boltzmann Superposition Principle	p. 377
9.2.4 Linearly Viscoelastic Material Models	p. 380
9.2.5 Beam Theory	p. 385
9.2.6 Torsion	p. 388
9.3 The Correspondence Principle	p. 388
9.3.1 Quasi-Static Problems	p. 391
9.4 Dynamic Response	p. 394
9.4.1 Complex Notation	p. 398
9.4.2 Viscoelastic Foundation	p. 404
9.5 Viscoelastic Waves	p. 407
9.5.1 Acceleration Waves in a Cylindrical Bar	p. 407
9.5.2 Progressive Harmonic Wave in a Cylindrical Bar	p. 411
9.5.3 Waves in Infinite Viscoelastic Medium	p. 414
9.6 Non-Linear Viscoelasticity	p. 419
9.6.1 The Norton Fluid	p. 422
9.6.2 Steady Bending of Non-Linearly Viscoelastic Beams	p. 423
Problems	p. 425
10 Theory of Plasticity	p. 433
10.1 Introduction	p. 433
10.2 Yield Criteria	p. 435
10.2.1 The Mises Yield Criterion	p. 440
10.2.2 The Tresca Yield Criterion	p. 444
10.2.3 Yield Criteria for Hardening Materials	p. 449
10.3 Flow Rules	p. 451
10.3.1 The General Flow Rule	p. 451
10.3.2 Elastic-Perfectly Plastic Tresca Material	p. 452
10.3.3 Elastic-Perfectly Plastic Mises Material	p. 457
10.4 Elastic-Plastic Analysis	p. 458
10.4.1 Plane Stress Problems	p. 459
10.4.2 Plane Strain Problems	p. 463
10.4.3 General Two-Dimensional Problem	p. 466
10.5 Limit Load Analysis for Plane Beams and Frames	p. 471
10.5.1 Introduction	p. 471
10.5.2 Plastic Collapse	p. 471
10.5.3 Limit Load Theorem for Plane Beams and Frames	p. 477
10.6 The Drucker Postulate	p. 479
10.7 Limit Load Analysis	p. 483
10.7.1 Lower Bound Limit Load Theorem	p. 485
10.7.2 Upper Bound Limit Load Theorem	p. 486
10.7.3 Discontinuity in Stress and Velocity	p. 489
10.7.4 Indentation	p. 491
10.8 Yield Line Theory	p. 495
10.9 Mises Material with Isotropic Hardening	p. 503
10.10 Yield Criteria Dependent on the Mean Stress	p. 507
10.10.1 The Mohr-Coulomb Criterion	p. 507
10.10.2 The Drucker-Prager Criterion	p. 510
10.11 Viscoplasticity	p. 511
10.11.1 Introduction	p. 511
10.11.2 The Bingham Elasto-Viscoplastic Models	p. 511
Problems	p. 515
11 Constitutive Equations	p. 517
11.1 Introduction	p. 517
11.2 Objective Tensor Fields	p. 519
11.2.1 Tensor Components in Two References	p. 521
11.2.2 Material Derivative of Objective Tensors	p. 522
11.2.3 Deformations with Respect to Fixed Reference Configuration	p. 524
11.2.4 Deformation with Respect to the Present Configuration	p. 527
11.3 Corotational Derivative	p. 530
11.4 Convected Derivative	p. 531
11.5 General Principles of Constitutive Theory	p. 532
11.5.1 Present Configuration as Reference Configuration	p. 536
11.6 Material Symmetry	p. 539
11.6.1 Symmetry Groups	p. 540
11.6.2 Isotropy	p. 542
11.6.3 Change of Reference Configuration	p. 543
11.6.4 Classification of Simple Materials	p. 544
11.6.5 Liquid Crystals	p. 548
11.7 Thermoelastic Materials	p. 548
11.8 Thermoviscous Fluids	p. 551
11.9 Advanced Fluid Models	p. 552
11.9.1 Introduction	p. 552
11.9.2 Stokesian Fluids or Reiner-Rivlin Fluids	p. 553
11.9.3 Corotational Fluid Models	p. 554
11.9.4 Quasi-Linear Corotational Fluid Models	p. 556
11.9.5 Oldroyd Fluids	p. 557
12 Tensors in Euclidean Space E[subscript 3]	p. 561
12.1 Introduction	p. 561
12.2 General Coordinates. Base Vectors	p. 561
12.2.1 Covariant and Contravariant Transformations	p. 564
12.2.2 Fundamental Parameters of a Coordinate System	p. 567
12.2.3 Orthogonal Coordinates	p. 568
12.3 Vector Fields	p. 569
12.4 Tensor Fields	p. 573
12.4.1 Tensor Components. Tensor Algebra	p. 573
12.4.2 Symmetric Tensors of 2. Order	p. 575
12.4.3 Tensors as Polyadics	p. 576
12.5 Differentiation of Tensors	p. 577
12.5.1 Christoffel Symbols	p. 577
12.5.2 Absolute and Covariant Derivatives of Vector Components	p. 578
12.5.3 The Frenet-Serret Formulas of Space Curves	p. 582
12.5.4 Divergence and Rotation of a Vector Field	p. 583
12.5.5 Orthogonal Coordinates	p. 584
12.5.6 Absolute and Covariant Derivatives of Tensor Components	p. 586
12.6 Integration of Tensor Fields	p. 591
12.7 Two-Point Tensor Components	p. 592
12.8 Relative Tensors	p. 595
Problems	p. 596
13 Continuum Mechanics in Curvilinear Coordinates	p. 599
13.1 Introduction	p. 599
13.2 Kinematics	p. 599
13.3 Deformation Analysis	p. 601
13.3.1 Strain Measures	p. 601
13.3.2 Small Strains and Small Deformations	p. 603
13.3.3 Rates of Deformation, Strain, and Rotation	p. 605
13.3.4 Orthogonal Coordinates	p. 605
13.3.5 General Analysis of Large Deformations	p. 607
13.3.6 Convected Coordinates	p. 608
13.4 Convected Derivatives of Tensors	p. 611
13.5 Stress Tensors. Equations of Motion	p. 615
13.5.1 Physical Stress Components	p. 615
13.5.2 Cauchy Equations of Motion	p. 617
13.6 Basic Equations in Elasticity	p. 618
13.7 Basic Equations in Fluid Mechanics	p. 619
13.7.1 Perfect Fluids [identical with] Eulerian Fluids	p. 620
13.7.2 Linearly Viscous Fluids [identical with] Newtonian Fluids	p. 620
13.7.3 Orthogonal Coordinates	p. 621
Problems	p. 623
Appendices	p. 625
Appendix A Del-Operator	p. 625
Appendix B The Navier - Stokes Equations	p. 626
Appendix C Integral Theorems	p. 627
References	p. 643
Symbols	p. 645
Index	p. 649

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