Nonlinear Problems of Elasticity

Preface vii
Chapter 1. Background 1
1. Notational and Terminological Conventions 1
2. Prerequisites 2
3. Functions 2
4. Vectors 4
5. Differential Equations 7
6. Notation for Sets 8
7. Real-Variable Theory 9
8. Function Spaces 10
Chapter 2. The Equations of Motion for
Extensible Strings 13
1. Introduction 13
2. The Classical Equations of Motion 14
3. The Linear Impulse-Momentum Law 25
4. The Equivalence of the Linear Impulse-Momentum
Law with the Principle of Virtual Power 28
5. Jump Conditions 31
6. The Existence of a Straight Equilibrium State 33
7. Purely Transverse Motions 35
8. Perturbation Methods and the LinearWave Equation 37
9. The Justification of Perturbation Methods 43
10. Variational Characterization of the Equations 46
11. Discretization 11
Chapter 3. ElementaryProblems forElastic Strings 53
1. Introduction 53
2. Equilibrium of Strings Under Vertical Loads 54
3. The Catenary Problem 57
4. The Suspension Bridge Problem 69
5. Equilibrium of Strings under Normal Loads 71
6. Equilibrium of Strings Under Central Forces 79
7. Circular States of a Spinning String 82
8. Travelling Waves 83
9. Radial Oscillations 85
10. Combined Whirling and Radial Motions 86
11. Massless Springs 86
xiii
xiv CONTENTS
12. Comments and Historical Notes 92
Chapter 4. Planar Steady-State Problems
for Elastic Rods 93
1. Formulation of the Governing Equations 93
2. Planar Equilibrium States of Straight Rods
under Terminal Loads 104
3. Equilibrium of Rings under Hydrostatic Pressure 109
4. Asymptotic Shape of Inflated Rings 118
5. Straight Configurations of a Whirling Rod 124
6. Simultaneous Whirling and Breathing Oscillations
of a Ring 131
7. Bibliographical Notes 132
Chapter 5. Introduction to Bifurcation Theory
and its Applications to Elasticity 133
1. The Simplest Buckling Problem 133
2. Classical Buckling Problems of Elasticity 139
3. Mathematical Concepts and Examples 146
4. Basic Theorems of Bifurcation Theory 157
5. Applications of the Basic Theorems
to the Simplest Buckling Problem 165
6. Perturbation Methods 171
7. Dynamics and Stability 176
8. Bibliographical Notes 179
Chapter 6. Global Bifurcation Problems
for Strings and Rods 179
1. The Equations for the Steady Whirling of Strings 179
2. Kolodner's Problem 182
3. Other Problems for Whirling Strings 191
4. The Drawing and Whirling of Strings 196
5. Planar Buckling of Rods. Global Theory 202
6. Planar Buckling of Rods. Imperfection Sensitivity
via Singularity Theory 205
7. Planar Buckling of Rods. Constitutive Assumptions 209
8. Planar Buckling of Rods. Nonbifurcating Branches 212
9. Global Disposition of Solution Sheets 214
10. Other Planar Buckling Problems for Straight Rods 219
11. Follower Loads 221
12. Buckling of Arches 223
13. Buckling of Whirling Rods 227
Chapter 7. Variational Methods 231
1. Introduction 231
2. The Multiplier Rule 234
3. Direct Methods 238
4. The Bootstrap Method 245
5. Inflation Problems 249
6. Problems for Whirling Rods 255
CONTENTS xv
7. The Second Variation. Bifurcation Problems 257
8. Notes 260
Chapter 8. Theory of Rods Deforming in Space 261
1. Introduction 261
2. Outline of the Essential Theory 262
3. The Exact Equations of Motion 268
4. The Equations of Constrained Motion 272
5. The Use of Exact Momenta 275
6. The Strains and the Strain-Rates 276
7. The Preservation of Orientation 279
8. Constitutive Equations Invariant
under Rigid Motions 284
9. Invariant Dissipative Mechanisms
for Numerical Schemes 289
10. Monotonicity and Growth Conditions 294
11. Transverse Hemitropy and Isotropy 300
12. Uniform Rods. Singular Problems 309
13. Representations for the Directors in Terms
of Euler Angles 311
14. Boundary Conditions 313
15. Impulse-Momentum Laws and the Principle
of Virtual Power 319
16. Hamilton's Principle for Hyperelastic Rods 323
17. Material Constraints 324
18. Planar Motions 327
19. Classical Theories 329
20. General Theories of Cosserat Rods 332
21. Historical and Bibliographical Notes 333
Chapter 9. Spatial Problems for Rods 335
1. Summary of the Governing Equations 335
2. Kirchhoff's Problem for Helical Equilibrium States 337
3. General Solutions for Equlibria 340
4. Travelling Waves in Straight Rods 344
5. Buckling under Terminal Thrust and Torque 347
6. Lateral Instability 349
Chapter 10. Axisymmetric Equilibria of Shells 353
1. Formulation of the Governing Equations 353
2. Buckling of a Transversely Isotropic Circular Plate 359
3. Remarkable Trivial States
of Aeolotropic Circular Plates 366
4. Buckling of Aeolotropic Plates 373
5. Buckling of Spherical Shells 375
6. Buckling of Cylindrical Shells 379
7. Asymptotic Shape of Inflated Shells 380
8. Membranes 381
9. Eversion 383
xvi CONTENTS
Chapter 11. Tensors 389
1. Tensor Algebra 389
2. Tensor Calculus 398
3. Indicial Notation 404
Chapter 12. 3-Dimensional Continuum
Mechanics 407
1. Kinematics 407
2. Strain 409
3. Compatibility 413
4. Rotation 416
5. Examples 418
6. Mass and Density 422
7. Stress and the Equations of Motion 423
8. Boundary and Initial Conditions 430
9. Impulse-Momentum Laws and
the Principle of Virtual Power 433
10. Constitutive Equations of Mechanics 439
11. Invariance under Rigid Motions 441
12. Material Constraints 445
13. Isotropy 458
14. Thermomechanics 463
15. The Spatial Formulation 473
Chapter 13. 3-Dimensional Theory of Nonlinear
Elasticity 479
1. Summary of the Governing Equations 479
2. Constitutive Restrictions 481
3. Order-Preservation: Monotonicity and Ellipticity 482
4. Growth Conditions 490
5. Special Constitutive Equations 491
6. Existence and Regularity 492
7. Versions of the Euler-Lagrange Equations 493
8. Linear Elasticity 497
9. Viscous Dissipation 499
Chapter 14. Problems in Nonlinear Elasticity 501
1. Elementary Static Problems in Cartesian Coordinates 501
2. Torsion, Extension, Inflation, and Shear
of an Annular Sector 503
3. Torsion and Related Equilibrium Problems
for Incompressible Bodies 507
4. Torsion, Extension, Inflation, and Shear
of a Compressible Annular Sector 511
5. Flexure, Extension, and Shear of a Block 514
6. Flexure, Extension, and Shear of a Compressible Block 516
7. Dilatation, Cavitation, Inflation, and Eversion 522
8. Other Semi-Inverse Problems 537
9. Universal and Non-Universal Deformations 540
CONTENTS xvii
10. Antiplane Problems 543
11. Perturbation Methods 545
12. Instability of an Incompressible Body under Constant
Normal Traction 556
13. Radial Motions of an Incompressible Tube 560
14. Universal Motions of Incompressible Bodies 562
15. Standing Shear Waves in an Incompressible Layer 567
16. Commentary. Other Problems 570
Chapter 15. Large-Strain Plasticity 571
1. Constitutive Equations 571
2. Refinements and Generalizations 577
3. Example: Longitudinal Motion of a Bar 579
4. Antiplane Shearing Motions 581
5. Notes 585
Chapter 16. General Theories of Rods 587
1. Introduction 587
2. Curvilinear Coordinates 589
3. Geometry of Rod-like Bodies 592
4. Exact Equations of Motion 593
5. Semi-Intrinsic Theories 594
6. Induced Theories of Rods 595
7. Convergence 604
8. Rods with Two Directors 607
9. Elastic Rods 615
10. Elastic Rods with a Plane of Symmetry 619
11. Necking 624
12. The Treatment of Incompressibility 628
13. Intrinsic Theories of Rods 636
14. Mielke's Treatment of St. Venant's Principle 638
Chapter 17. General Theories of Shells 643
1. Induced Shell Theories 643
2. Shells with One Director 647
3. Drawing and Twisting of an Elastic Plate 653
4. Axisymmetric Motions of Axisymmetric Shells 657
5. Global Buckled States of a Cosserat Plate 663
6. Thickness parameter. Eversion 666
7. The Treatment of Incompressibility 668
8. Intrinsic Theory of Special Cosserat Shells 669
9. Membranes 680
10. Asymptotic Methods. The von Kïarmïan Equations 682
11. Justification of Shell Theories as Asymptotic Limits 687
12. Commentary. Historical Notes 690
Chapter 18. Dynamical Problems 693
1. The 1-Dimensional Quasilinear Wave Equation 693
2. The Riemann Problem. Uniqueness and
Admissibility of Weak Solutions 697
xviii CONTENTS
3. Shearing Motions of Viscoplastic Layers 704
4. Dissipative Mechanisms and the Bounds They Induce 705
5. Shock Structure. Admissibility and Travelling Waves 715
6. Travelling Shear Waves in Viscoelatic Media 720
7. Blow-up in Three-Dimensional Hyperelasticity 727
Chapter 19. Appendix. Topics in Linear Analysis 733
1. Banach Spaces 733
2. Linear Operators and Linear Equations 736
Chapter 20. Appendix. Local Nonlinear Analysis 741
1. The Contraction Mapping Principle and the Implicit
Function Theorem 741
2. The Lyapunov-Schmidt Method. The
Poincarïe Shooting Method 744
Chapter 21. Appendix. Degree Theory and its
Applications 747
1. Definition of the Brouwer Degree 747
2. Properties of the Brouwer Degree 751
3. Leray-Schauder Degree 757
4. One-Parameter Global Bifurcation Theorem 761
References 763
Index 801