Classical And Computational Solid Mechanics

By: Pin Tong, Yuen-cheng Fung (Editor)

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Paperback | 10 July 2001

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952 Pages

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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.

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Introduction	p. 1
Hooke's Law	p. 2
Linear Solids with Memory	p. 9
Sinusoidal Oscillations in Viscoelastic Material: Models of Viscoelasticity	p. 12
Plasticity	p. 14
Vibrations	p. 15
Prototype of Wave Dynamics	p. 18
Biomechanics	p. 22
Historical Remarks	p. 25
Tensor Analysis	p. 30
Notation and Summation Convention	p. 30
Coordinate Transformation	p. 33
Euclidean Metric Tensor	p. 34
Scalars, Contravariant Vectors, Covariant Vectors	p. 38
Tensor Fields of Higher Rank	p. 39
Some Important Special Tensors	p. 40
The Significance of Tensor Characteristics	p. 42
Rectangular Cartesian Tensors	p. 43
Contraction	p. 44
Quotient Rule	p. 45
Partial Derivatives in Cartesian Coordinates	p. 46
Covariant Differentiation of Vector Fields	p. 48
Tensor Equations	p. 49
Geometric Interpretation of Tensor Components	p. 52
Geometric Interpretation of Covariant Derivatives	p. 58
Physical Components of a Vector	p. 60
Stress Tensor	p. 66
Stresses	p. 66
Laws of Motion	p. 69
Cauchy's Formula	p. 71
Equations of Equilibrium	p. 73
Transformation of Coordinates	p. 78
Plane State of Stress	p. 79
Principal Stresses	p. 82
Shearing Stresses	p. 85
Mohr's Circles	p. 86
Stress Deviations	p. 87
Octahedral Shearing Stress	p. 88
Stress Tensor in General Coordinates	p. 90
Physical Components of a Stress Tensor in General Coordinates	p. 94
Equations of Equilibrium in Curvilinear Coordinates	p. 95
Analysis of Strain	p. 97
Deformation	p. 97
Strain Tensors in Rectangular Cartesian Coordinates	p. 100
Geometric Interpretation of Infinitesimal Strain Components	p. 103
Rotation	p. 104
Finite Strain Components	p. 106
Compatibility of Strain Components	p. 108
Multiply Connected Regions	p. 113
Multivalued Displacements	p. 117
Properties of the Strain Tensor	p. 118
Physical Components	p. 121
Example--Spherical Coordinates	p. 123
Example--Cylindrical Polar Coordinates	p. 125
Conservation Laws	p. 127
Gauss' Theorem	p. 127
Material and Spatial Descriptions of Changing Configurations	p. 128
Material Derivative of Volume Integral	p. 131
The Equation of Continuity	p. 133
Equations of Motion	p. 134
Moment of Momentum	p. 135
Other Field Equations	p. 136
Elastic and Plastic Behavior of Materials	p. 138
Generalized Hooke's Law	p. 138
Stress-Strain Relationship for Isotropic Elastic Materials	p. 140
Ideal Plastic Solids	p. 143
Some Experimental Information	p. 146
A Basic Assumption of the Mathematical Theory of Plasticity	p. 150
Loading and Unloading Criteria	p. 156
Isotropic Stress Theories of Yield Function	p. 157
Further Examples of Yield Functions	p. 159
Work Hardening--Drucker's Hypothesis and Definition	p. 166
Ideal Plasticity	p. 167
Flow Rule for Work-Hardening Materials	p. 171
Subsequent Loading Surfaces--Isotropic and Kinematic Hardening Rules	p. 177
Mroz's, Dafalias and Popov's, and Valanis' Plasticity Theories	p. 189
Strain Space Formulations	p. 195
Finite Deformation	p. 199
Plastic Deformation of Crystals	p. 200
Linearized Theory of Elasticity	p. 203
Basic Equations of Elasticity for Homogeneous Isotropic Bodies	p. 203
Equilibrium of an Elastic Body Under Zero Body Force	p. 206
Boundary Value Problems	p. 207
Equilibrium and Uniqueness of Solutions	p. 210
Saint Venant's Theory of Torsion	p. 213
Soap Film Analogy	p. 222
Bending of Beams	p. 224
Plane Elastic Waves	p. 229
Rayleigh Surface Wave	p. 231
Love Wave	p. 235
Solutions of Problems in Linearized Theory of Elasticity by Potentials	p. 238
Scalar and Vector Potentials for Displacement Vector Fields	p. 238
Equations of Motion in Terms of Displacement Potentials	p. 241
Strain Potential	p. 243
Galerkin Vector	p. 246
Equivalent Galerkin Vectors	p. 249
Example--Vertical Load on the Horizontal Surface of a Semi-Infinite Solid	p. 250
Love's Strain Function	p. 252
Kelvin's Problem--A Single Force Acting in the Interior of an Infinite Solid	p. 254
Perturbation of Elasticity Solutions by a Change of Poisson's Ratio	p. 259
Boussinesq's Problem	p. 262
On Biharmonic Functions	p. 263
Neuber-Papkovich Representation	p. 268
Other Methods of Solution of Elastostatic Problems	p. 270
Reflection and Refraction of Plane P and S Waves	p. 270
Lamb's Problem--Line Load Suddenly Applied on Elastic Half-Space	p. 273
Two-Dimensional Problems in Linearized Theory of Elasticity	p. 280
Plane State of Stress or Strain	p. 280
Airy Stress Functions for Two-Dimensional Problems	p. 282
Airy Stress Function in Polar Coordinates	p. 288
General Case	p. 295
Representation of Two-Dimensional Biharmonic Functions by Analytic Functions of a Complex Variable	p. 299
Kolosoff-Muskhelishvili Method	p. 301
Variational Calculus, Energy Theorems, Saint-Venant's Principle	p. 313
Minimization of Functionals	p. 313
Functional Involving Higher Derivatives of the Dependent Variable	p. 319
Several Unknown Functions	p. 320
Several Independent Variables	p. 323
Subsidiary Conditions--Lagrangian Multipliers	p. 325
Natural Boundary Conditions	p. 328
Theorem of Minimum Potential Energy Under Small Variations of Displacements	p. 330
Example of Application: Static Loading on a Beam--Natural and Rigid End Conditions	p. 335
The Complementary Energy Theorem Under Small Variations of Stresses	p. 339
Variational Functionals Frequently Used in Computational Mechanics	p. 346
Saint-Venant's Principle	p. 355
Saint-Venant's Principle-Boussinesq-Von Mises-Sternberg Formulation	p. 359
Practical Applications of Saint-Venant's Principle	p. 362
Extremum Principles for Plasticity	p. 365
Limit Analysis	p. 369
Hamilton's Principle, Wave Propagation, Applications of Generalized Coordinates	p. 379
Hamilton's Principle	p. 379
Example of Application--Equation of Vibration of a Beam	p. 383
Group Velocity	p. 393
Hopkinson's Experiment	p. 396
Generalized Coordinates	p. 398
Approximate Representation of Functions	p. 399
Approximate Solution of Differential Equations	p. 402
Direct Methods of Variational Calculus	p. 402
Elasticity and Thermodynamics	p. 407
The Laws of Thermodynamics	p. 407
The Energy Equation	p. 412
The Strain Energy Function	p. 414
The Conditions of Thermodynamic Equilibrium	p. 416
The Positive Definiteness of the Strain Energy Function	p. 418
Thermodynamic Restrictions on the Stress-Strain Law of an Isotropic Elastic Material	p. 419
Generalized Hooke's Law, Including the Effect of Thermal Expansion	p. 421
Thermodynamic Functions for Isotropic Hookean Materials	p. 423
Equations Connecting Thermal and Mechanical Properties of a Solid	p. 425
Irreversible Thermodynamics and Viscoelasticity	p. 428
Basic Assumptions	p. 428
One-Dimensional Heat Conduction	p. 431
Phenomenological Relations-Onsager Principle	p. 432
Basic Equations of Thermomechanics	p. 436
Equations of Evolution for a Linear Hereditary Material	p. 440
Relaxation Modes	p. 444
Normal Coordinates	p. 447
Hidden Variables and the Force-Displacement Relationship	p. 450
Anisotropic Linear Viscoelastic Materials	p. 454
Thermoelasticity	p. 456
Basic Equations	p. 456
Thermal Effects Due to a Change of Strain; Kelvin's Formula	p. 459
Ratio of Adiabatic to Isothermal Elastic Moduli	p. 459
Uncoupled, Quasi-Static Thermoelastic Theory	p. 461
Temperature Distribution	p. 462
Thermal Stresses	p. 464
Particular Integral: Goodier's Method	p. 466
Plane Strain	p. 467
An Example--Stresses in a Turbine Disk	p. 470
Variational Principle for Uncoupled Thermoelasticity	p. 473
Variational Principle for Heat Conduction	p. 474
Coupled Thermoelasticity	p. 478
Lagrangian Equations for Heat Conduction and Thermoelasticity	p. 481
Viscoelasticity	p. 487
Viscoelastic Material	p. 487
Stress-Strain Relations in Differential Equation Form	p. 491
Boundary-Value Problems and Integral Transformations	p. 497
Waves in an Infinite Medium	p. 500
Quasi-Static Problems	p. 503
Reciprocity Relations	p. 507
Large Deformation	p. 514
Coordinate Systems and Tensor Notation	p. 514
Deformation Gradient	p. 521
Strains	p. 525
Right and Left Stretch Strain and Rotation Tensors	p. 526
Strain Rates	p. 528
Material Derivatives of Line, Area, and Volume Elements	p. 529
Stresses	p. 532
Example: Combined Tension and Torsion Loads	p. 539
Objectivity	p. 543
Equations of Motion	p. 548
Constitutive Equations of Thermoelastic Bodies	p. 550
More Examples	p. 557
Variational Principles for Finite Elasticity: Compressible Materials	p. 562
Variational Principles for Finite Elasticity: Nearly Incompressible or Incompressible Materials	p. 568
Small Deflection of Thin Plates	p. 573
Large Deflection of Plates	p. 581
Incremental Approach to Solving Some Nonlinear Problems	p. 587
Updated Lagrangian Description	p. 587
Linearized Rates of Deformation	p. 590
Linearized Rates of Stress Measures	p. 593
Incremental Equations of Motion	p. 597
Constitutive Laws	p. 598
Incremental Variational Principles in Terms of T	p. 604
Incremental Variational Principles in Terms of r	p. 610
Incompressible and Nearly Incompressible Materials	p. 612
Updated Solution	p. 617
Incremental Loads	p. 620
Infinitesimal Strain Theory	p. 622
Finite Element Methods	p. 624
Basic Approach	p. 626
One Dimensional Problems Governed by a Second Order Differential Equation	p. 629
Shape Functions and Element Matrices for Higher Order Ordinary Differential Equations	p. 638
Assembling and Constraining Global Matrices	p. 643
Equation Solving	p. 651
Two Dimensional Problems by One-Dimensional Elements	p. 655
General Finite Element Formulation	p. 657
Convergence	p. 664
Two-Dimensional Shape Functions	p. 665
Element Matrices for a Second-Order Elliptical Equation	p. 672
Coordinate Transformation	p. 676
Triangular Elements with Curved Sides	p. 679
Quadrilateral Elements	p. 682
Plane Elasticity	p. 690
Three-Dimensional Shape Functions	p. 702
Three Dimensional Elasticity	p. 708
Dynamic Problems of Elastic Solids	p. 714
Numerical Integration	p. 726
Patch Tests	p. 731
Locking-Free Elements	p. 735
Spurious Modes in Reduced Integration	p. 750
Perspective	p. 754
Mixed and Hybrid Formulations	p. 756
Mixed Formulations	p. 756
Hybrid Formulations	p. 760
Hybrid Singular Elements (Super-Elements)	p. 767
Elements for Heterogeneous Materials	p. 782
Elements for Infinite Domain	p. 782
Incompressible or Nearly Incompressible Elasticity	p. 788
Finite Element Methods for Plates and Shells	p. 795
Linearized Bending Theory of Thin Plates	p. 795
Reissner-Mindlin Plates	p. 805
Mixed Functionals for Reissner Plate Theory	p. 813
Hybrid Formulations for Plates	p. 819
Shell as an Assembly of Plate Elements	p. 822
General Shell Elements	p. 832
Locking and Stabilization in Shell Applications	p. 843
Finite Element Modeling of Nonlinear Elasticity, Viscoelasticity, Plasticity, Viscoplasticity and Creep	p. 848
Updated Lagrangian Solution for Large Deformation	p. 849
Incremental Solution	p. 852
Dynamic Solution	p. 854
Newton-Raphson Iteration Method	p. 855
Viscoelasticity	p. 857
Plasticity	p. 859
Viscoplasticity	p. 869
Creep	p. 870
Bibliography	p. 873
Author Index	p. 909
Subject Index	p. 919
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Classical And Computational Solid Mechanics

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New Edition

Paperback

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