Classical and computational solid mechanics

Title:

Personal Author:

Fung, Y. C. (Yuan-cheng), 1919-

Publication Information:

River Edge, N.J. : World Scientific Publishing, 2001

ISBN:

9789810239121

Subject Term:

Mechanics, Applied -- Mathematical models

Engineering mathematics

Added Author:

Tong, Pin

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This invaluable book has been written for engineers and engineering scientists in a style that is readable, precise, concise, and practical. It gives first priority to the formulation of problems, presenting the classical results as the gold standard, and the numerical approach as a tool for obtaining solutions. The classical part is a revision of the well-known text Foundations of Solid Mechanics, with a much-expanded discussion on the theories of plasticity and large elastic deformation with finite strains. The computational part is all new and is aimed at solving many major linear and nonlinear boundary-value problems.

Author Notes

Y. C. Fung works on solid mechanics, and has contributed to expand its frontiers bordering aerodynamics and biology. He helped to establish the fields of aeroelasticity and biomechanics. He received his BS and MS from the National Central University in China and PhD from the California Institute of Technology. He is the recipient of the National Medal of Science from the President of USA, the Founders Award of the National Academy of Engineering of USA, the von Karman Medal from ASCE. Timoshenko Medal from ASME. Poiseuille Medal from ISB. Borelli Medal from ASB. Landis Award from Microcirculatory Society, and Alza Award from BMES. He is a member of the US National Academy of Science. National Academy of Engineering. Institute of Medicine of NAS. Academia Sinica, and Chinese Academy of Science. He is Honorary Professor of 16 universities in China, and Distinguished Alumnus of Caltech
Pin Tong is a preeminent developer of computational solid mechanics. He uses more and more general variational principles that admit less and less restrictive hypotheses. He received his BS degree from Taiwan University, and his MS and PhD from the California Institute of Technology. He was Research Fellow at Caltech. Assistant and Associate Professor of Aeronautics and Astronautics at MIT. Chief of Structures and Dynamics Division of US Department of Transportation. Transportation Systems Center. He is Professor of Mechanical Engineering and Founding Head of the Department at the Hong Kong University of Science and Technology. Author of over 100 papers and a book on Finite Element Method. Dr Tong is regional editor of the Int J of Fracture, and Int J Comp Mech, and President of Far East Oceanic Fracture Soc, HK Soc Theoretical and Applied Mechanics, and HK Sec ASME. Dr Tong received the von Karman Award for his outstanding contribution to structural materials, the Engineer of the Year Award, and the Award for Meritorious Achievement from the US Department of Transportation

1 Introduction	p. 1
1.1. Hooke's Law	p. 2
1.2. Linear Solids with Memory	p. 9
1.3. Sinusoidal Oscillations in Viscoelastic Material: Models of Viscoelasticity	p. 12
1.4. Plasticity	p. 14
1.5. Vibrations	p. 15
1.6. Prototype of Wave Dynamics	p. 18
1.7. Biomechanics	p. 22
1.8. Historical Remarks	p. 25
2 Tensor Analysis	p. 30
2.1. Notation and Summation Convention	p. 30
2.2. Coordinate Transformation	p. 33
2.3. Euclidean Metric Tensor	p. 34
2.4. Scalars, Contravariant Vectors, Covariant Vectors	p. 38
2.5. Tensor Fields of Higher Rank	p. 39
2.6. Some Important Special Tensors	p. 40
2.7. The Significance of Tensor Characteristics	p. 42
2.8. Rectangular Cartesian Tensors	p. 43
2.9. Contraction	p. 44
2.10. Quotient Rule	p. 45
2.11. Partial Derivatives in Cartesian Coordinates	p. 46
2.12. Covariant Differentiation of Vector Fields	p. 48
2.13. Tensor Equations	p. 49
2.14. Geometric Interpretation of Tensor Components	p. 52
2.15. Geometric Interpretation of Covariant Derivatives	p. 58
2.16. Physical Components of a Vector	p. 60
3 Stress Tensor	p. 66
3.1. Stresses	p. 66
3.2. Laws of Motion	p. 69
3.3. Cauchy's Formula	p. 71
3.4. Equations of Equilibrium	p. 73
3.5. Transformation of Coordinates	p. 78
3.6. Plane State of Stress	p. 79
3.7. Principal Stresses	p. 82
3.8. Shearing Stresses	p. 85
3.9. Mohr's Circles	p. 86
3.10. Stress Deviations	p. 87
3.11. Octahedral Shearing Stress	p. 88
3.12. Stress Tensor in General Coordinates	p. 90
3.13. Physical Components of a Stress Tensor in General Coordinates	p. 94
3.14. Equations of Equilibrium in Curvilinear Coordinates	p. 95
4 Analysis of Strain	p. 97
4.1. Deformation	p. 97
4.2. Strain Tensors in Rectangular Cartesian Coordinates	p. 100
4.3. Geometric Interpretation of Infinitesimal Strain Components	p. 103
4.4. Rotation	p. 104
4.5. Finite Strain Components	p. 106
4.6. Compatibility of Strain Components	p. 108
4.7. Multiply Connected Regions	p. 113
4.8. Multivalued Displacements	p. 117
4.9. Properties of the Strain Tensor	p. 118
4.10. Physical Components	p. 121
4.11. Example--Spherical Coordinates	p. 123
4.12. Example--Cylindrical Polar Coordinates	p. 125
5 Conservation Laws	p. 127
5.1. Gauss' Theorem	p. 127
5.2. Material and Spatial Descriptions of Changing Configurations	p. 128
5.3. Material Derivative of Volume Integral	p. 131
5.4. The Equation of Continuity	p. 133
5.5. Equations of Motion	p. 134
5.6. Moment of Momentum	p. 135
5.7. Other Field Equations	p. 136
6 Elastic and Plastic Behavior of Materials	p. 138
6.1. Generalized Hooke's Law	p. 138
6.2. Stress-Strain Relationship for Isotropic Elastic Materials	p. 140
6.3. Ideal Plastic Solids	p. 143
6.4. Some Experimental Information	p. 146
6.5. A Basic Assumption of the Mathematical Theory of Plasticity	p. 150
6.6. Loading and Unloading Criteria	p. 156
6.7. Isotropic Stress Theories of Yield Function	p. 157
6.8. Further Examples of Yield Functions	p. 159
6.9. Work Hardening--Drucker's Hypothesis and Definition	p. 166
6.10. Ideal Plasticity	p. 167
6.11. Flow Rule for Work-Hardening Materials	p. 171
6.12. Subsequent Loading Surfaces--Isotropic and Kinematic Hardening Rules	p. 177
6.13. Mroz's, Dafalias and Popov's, and Valanis' Plasticity Theories	p. 189
6.14. Strain Space Formulations	p. 195
6.15. Finite Deformation	p. 199
6.16. Plastic Deformation of Crystals	p. 200
7 Linearized Theory of Elasticity	p. 203
7.1. Basic Equations of Elasticity for Homogeneous Isotropic Bodies	p. 203
7.2. Equilibrium of an Elastic Body Under Zero Body Force	p. 206
7.3. Boundary Value Problems	p. 207
7.4. Equilibrium and Uniqueness of Solutions	p. 210
7.5. Saint Venant's Theory of Torsion	p. 213
7.6. Soap Film Analogy	p. 222
7.7. Bending of Beams	p. 224
7.8. Plane Elastic Waves	p. 229
7.9. Rayleigh Surface Wave	p. 231
7.10. Love Wave	p. 235
8 Solutions of Problems in Linearized Theory of Elasticity by Potentials	p. 238
8.1. Scalar and Vector Potentials for Displacement Vector Fields	p. 238
8.2. Equations of Motion in Terms of Displacement Potentials	p. 241
8.3. Strain Potential	p. 243
8.4. Galerkin Vector	p. 246
8.5. Equivalent Galerkin Vectors	p. 249
8.6. Example--Vertical Load on the Horizontal Surface of a Semi-Infinite Solid	p. 250
8.7. Love's Strain Function	p. 252
8.8. Kelvin's Problem--A Single Force Acting in the Interior of an Infinite Solid	p. 254
8.9. Perturbation of Elasticity Solutions by a Change of Poisson's Ratio	p. 259
8.10. Boussinesq's Problem	p. 262
8.11. On Biharmonic Functions	p. 263
8.12. Neuber-Papkovich Representation	p. 268
8.13. Other Methods of Solution of Elastostatic Problems	p. 270
8.14. Reflection and Refraction of Plane P and S Waves	p. 270
8.15. Lamb's Problem--Line Load Suddenly Applied on Elastic Half-Space	p. 273
9 Two-Dimensional Problems in Linearized Theory of Elasticity	p. 280
9.1. Plane State of Stress or Strain	p. 280
9.2. Airy Stress Functions for Two-Dimensional Problems	p. 282
9.3. Airy Stress Function in Polar Coordinates	p. 288
9.4. General Case	p. 295
9.5. Representation of Two-Dimensional Biharmonic Functions by Analytic Functions of a Complex Variable	p. 299
9.6. Kolosoff-Muskhelishvili Method	p. 301
10 Variational Calculus, Energy Theorems, Saint-Venant's Principle	p. 313
10.1. Minimization of Functionals	p. 313
10.2. Functional Involving Higher Derivatives of the Dependent Variable	p. 319
10.3. Several Unknown Functions	p. 320
10.4. Several Independent Variables	p. 323
10.5. Subsidiary Conditions--Lagrangian Multipliers	p. 325
10.6. Natural Boundary Conditions	p. 328
10.7. Theorem of Minimum Potential Energy Under Small Variations of Displacements	p. 330
10.8. Example of Application: Static Loading on a Beam--Natural and Rigid End Conditions	p. 335
10.9. The Complementary Energy Theorem Under Small Variations of Stresses	p. 339
10.10. Variational Functionals Frequently Used in Computational Mechanics	p. 346
10.11. Saint-Venant's Principle	p. 355
10.12. Saint-Venant's Principle-Boussinesq-Von Mises-Sternberg Formulation	p. 359
10.13. Practical Applications of Saint-Venant's Principle	p. 362
10.14. Extremum Principles for Plasticity	p. 365
10.15. Limit Analysis	p. 369
11 Hamilton's Principle, Wave Propagation, Applications of Generalized Coordinates	p. 379
11.1. Hamilton's Principle	p. 379
11.2. Example of Application--Equation of Vibration of a Beam	p. 383
11.3. Group Velocity	p. 393
11.4. Hopkinson's Experiment	p. 396
11.5. Generalized Coordinates	p. 398
11.6. Approximate Representation of Functions	p. 399
11.7. Approximate Solution of Differential Equations	p. 402
11.8. Direct Methods of Variational Calculus	p. 402
12 Elasticity and Thermodynamics	p. 407
12.1. The Laws of Thermodynamics	p. 407
12.2. The Energy Equation	p. 412
12.3. The Strain Energy Function	p. 414
12.4. The Conditions of Thermodynamic Equilibrium	p. 416
12.5. The Positive Definiteness of the Strain Energy Function	p. 418
12.6. Thermodynamic Restrictions on the Stress-Strain Law of an Isotropic Elastic Material	p. 419
12.7. Generalized Hooke's Law, Including the Effect of Thermal Expansion	p. 421
12.8. Thermodynamic Functions for Isotropic Hookean Materials	p. 423
12.9. Equations Connecting Thermal and Mechanical Properties of a Solid	p. 425
13 Irreversible Thermodynamics and Viscoelasticity	p. 428
13.1. Basic Assumptions	p. 428
13.2. One-Dimensional Heat Conduction	p. 431
13.3. Phenomenological Relations-Onsager Principle	p. 432
13.4. Basic Equations of Thermomechanics	p. 436
13.5. Equations of Evolution for a Linear Hereditary Material	p. 440
13.6. Relaxation Modes	p. 444
13.7. Normal Coordinates	p. 447
13.8. Hidden Variables and the Force-Displacement Relationship	p. 450
13.9. Anisotropic Linear Viscoelastic Materials	p. 454
14 Thermoelasticity	p. 456
14.1. Basic Equations	p. 456
14.2. Thermal Effects Due to a Change of Strain; Kelvin's Formula	p. 459
14.3. Ratio of Adiabatic to Isothermal Elastic Moduli	p. 459
14.4. Uncoupled, Quasi-Static Thermoelastic Theory	p. 461
14.5. Temperature Distribution	p. 462
14.6. Thermal Stresses	p. 464
14.7. Particular Integral: Goodier's Method	p. 466
14.8. Plane Strain	p. 467
14.9. An Example--Stresses in a Turbine Disk	p. 470
14.10. Variational Principle for Uncoupled Thermoelasticity	p. 473
14.11. Variational Principle for Heat Conduction	p. 474
14.12. Coupled Thermoelasticity	p. 478
14.13. Lagrangian Equations for Heat Conduction and Thermoelasticity	p. 481
15 Viscoelasticity	p. 487
15.1. Viscoelastic Material	p. 487
15.2. Stress-Strain Relations in Differential Equation Form	p. 491
15.3. Boundary-Value Problems and Integral Transformations	p. 497
15.4. Waves in an Infinite Medium	p. 500
15.5. Quasi-Static Problems	p. 503
15.6. Reciprocity Relations	p. 507
16 Large Deformation	p. 514
16.1. Coordinate Systems and Tensor Notation	p. 514
16.2. Deformation Gradient	p. 521
16.3. Strains	p. 525
16.4. Right and Left Stretch Strain and Rotation Tensors	p. 526
16.5. Strain Rates	p. 528
16.6. Material Derivatives of Line, Area, and Volume Elements	p. 529
16.7. Stresses	p. 532
16.8. Example: Combined Tension and Torsion Loads	p. 539
16.9. Objectivity	p. 543
16.10. Equations of Motion	p. 548
16.11. Constitutive Equations of Thermoelastic Bodies	p. 550
16.12. More Examples	p. 557
16.13. Variational Principles for Finite Elasticity: Compressible Materials	p. 562
16.14. Variational Principles for Finite Elasticity: Nearly Incompressible or Incompressible Materials	p. 568
16.15. Small Deflection of Thin Plates	p. 573
16.16. Large Deflection of Plates	p. 581
17 Incremental Approach to Solving Some Nonlinear Problems	p. 587
17.1. Updated Lagrangian Description	p. 587
17.2. Linearized Rates of Deformation	p. 590
17.3. Linearized Rates of Stress Measures	p. 593
17.4. Incremental Equations of Motion	p. 597
17.5. Constitutive Laws	p. 598
17.6. Incremental Variational Principles in Terms of T	p. 604
17.7. Incremental Variational Principles in Terms of r	p. 610
17.8. Incompressible and Nearly Incompressible Materials	p. 612
17.9. Updated Solution	p. 617
17.10. Incremental Loads	p. 620
17.11. Infinitesimal Strain Theory	p. 622
18 Finite Element Methods	p. 624
18.1. Basic Approach	p. 626
18.2. One Dimensional Problems Governed by a Second Order Differential Equation	p. 629
18.3. Shape Functions and Element Matrices for Higher Order Ordinary Differential Equations	p. 638
18.4. Assembling and Constraining Global Matrices	p. 643
18.5. Equation Solving	p. 651
18.6. Two Dimensional Problems by One-Dimensional Elements	p. 655
18.7. General Finite Element Formulation	p. 657
18.8. Convergence	p. 664
18.9. Two-Dimensional Shape Functions	p. 665
18.10. Element Matrices for a Second-Order Elliptical Equation	p. 672
18.11. Coordinate Transformation	p. 676
18.12. Triangular Elements with Curved Sides	p. 679
18.13. Quadrilateral Elements	p. 682
18.14. Plane Elasticity	p. 690
18.15. Three-Dimensional Shape Functions	p. 702
18.16. Three Dimensional Elasticity	p. 708
18.17. Dynamic Problems of Elastic Solids	p. 714
18.18. Numerical Integration	p. 726
18.19. Patch Tests	p. 731
18.20. Locking-Free Elements	p. 735
18.21. Spurious Modes in Reduced Integration	p. 750
18.22. Perspective	p. 754
19 Mixed and Hybrid Formulations	p. 756
19.1. Mixed Formulations	p. 756
19.2. Hybrid Formulations	p. 760
19.3. Hybrid Singular Elements (Super-Elements)	p. 767
19.4. Elements for Heterogeneous Materials	p. 782
19.5. Elements for Infinite Domain	p. 782
19.6. Incompressible or Nearly Incompressible Elasticity	p. 788
20 Finite Element Methods for Plates and Shells	p. 795
20.1. Linearized Bending Theory of Thin Plates	p. 795
20.2. Reissner-Mindlin Plates	p. 805
20.3. Mixed Functionals for Reissner Plate Theory	p. 813
20.4. Hybrid Formulations for Plates	p. 819
20.5. Shell as an Assembly of Plate Elements	p. 822
20.6. General Shell Elements	p. 832
20.7. Locking and Stabilization in Shell Applications	p. 843
21 Finite Element Modeling of Nonlinear Elasticity, Viscoelasticity, Plasticity, Viscoplasticity and Creep	p. 848
21.1. Updated Lagrangian Solution for Large Deformation	p. 849
21.2. Incremental Solution	p. 852
21.3. Dynamic Solution	p. 854
21.4. Newton-Raphson Iteration Method	p. 855
21.5. Viscoelasticity	p. 857
21.6. Plasticity	p. 859
21.7. Viscoplasticity	p. 869
21.8. Creep	p. 870
Bibliography	p. 873
Author Index	p. 909
Subject Index	p. 919

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