Stuart S. Antman Nonlinear Problems of Elasticity With 105 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest
Contents Preface vn Chapter I. Background 1 1. Notational and Terminological Conventions 1 2. Prerequisites 1 3. Functions 2 4. Vectors : 4 5. Differential Equations 6 6. Notation for Sets 7 7. Real Analysis 8 8. Function Spaces 9 Chapter II. The Equations of Motion for Extensible Strings 11 1. Introduction 11 2. The Classical Equations of Motion 12 3. The Linear Impulse-Momentum Law 22 4. The Equivalence of the Linear Impulse-Momentum Law with the Principle of Virtual Power 24 5. Jump Conditions 28 6. The Existence of a Straight Equilibrium State 29 7. Purely Transverse Motions 32 8. Perturbation Methods and the Linear Wave Equation 33 9. The Justification of Perturbation Methods \ 39 10. Variational Characterization of the Equations for an Elastic String 42 11. Discretization 46 Chapter III. Elementary Problems for Elastic Strings 49 1. Introduction 49 2. Equilibrium of Strings under Vertical Loads 50 3. The Catenary Problem 53 4. The Suspension Bridge Problem 64 5. Equilibrium of Strings under Normal Loads 66 6. Equilibrium of Strings under Central Forces 74 7. Dynamical Problems 77 8. Comments and Historical Notes 83
xiv CONTENTS Chapter IV. Planar Equilibrium Problems for Elastic Rods 85 1. Formulation of the Governing Equations 85 2. Planar Equilibrium States of Straight Rods under Terminal Loads 96 3. Equilibrium of Rings under Hydrostatic Pressure 101 4. Asymptotic Shape of Inflated Rings 110 5. Straight Configurations of a Whirling Rod 116 6. Bibliographical Notes 123 Chapter V. Introduction to Bifurcation Theory and its Applications to Elasticity 125 1. The Simplest Buckling Problem 125 2. Classical Buckling Problems of Elasticity 131 3. Mathematical Concepts and Examples 140 4. Basic Theorems of Bifurcation Theory 151 5. Applications of the Basic Theorems to the Simplest Buckling Problem 160 6. Perturbation Methods 166 7. Dynamics and Stability 170 8. Bibliographical Notes 172 Chapter VI. Global Bifurcation Problems for Strings and Rods 173 1. The Equations for the Steady Whirling of Strings 173 2. Kolodner's Problem 176 3. Other Problems for Whirling Strings. Bibliographical Notes... 185 4. The Drawing and Whirling of Strings 191 5. Planar Buckling of Rods. Global Theory 196 6. Planar Buckling of Rods. Imperfection Sensitivity via Singularity Theory. 200 7. Planar Buckling of Rods. Constitutive Assumptions.\... 204 8. Planar Buckling of Rods. Nonbifurcating Branches 207 9. Global Disposition of Solution Sheets 208 10. Other Planar Buckling Problems for Straight Rods 215 11. Follower Loads 217 12. Buckling of Arches.218 13. Buckling of Whirling Rods.222 Chapter VII. Variational Methods 227 1. Introduction 227 2. The Multiplier Rule 230 3. Direct Methods of the Calculus of Variations 234 4. The Bootstrap Method 242 5. Inflation Problems 245 6. Problems for Whirling Rods 251 7. The Second Variation. Bifurcation Problems 253
CONTENTS xv 8. Notes 256 Chapter VIII. The Special Cosserat Theory of Rods 259 1. Introduction 259 2. Outline of the Essential Theory 260 3. The Exact Equations of Motion 265 4. The Equations of Constrained Motion 269 5. The Strains and the Strain Rates 272 6. The Preservation of Orientation 276 7. Constitutive Equations Invariant under Rigid Motions 280 8. Monotonicity and Growth Conditions 286 9. Transverse Isotropy 292 10. Uniform Rods. Singular Problems 298 11. Representations for the Directors in Terms of Euler Angles... 300 12. Boundary Conditions 301 13. Impulse-Momentum Laws and the Principle of Virtual Power.. 308 14. Hamilton's Principle for Hyperelastic Rods 312 15. Material Constraints 314 16. External Constraints, Planar Problems, Classical Theories 317 17. General Theories of Cosserat Rods 322 18. Historical and Bibliographical Notes 323 Chapter IX. Spatial Problems for Cosserat Rods 325 1. Summary of the Governing Equations 325 2. Kirchhoff 's Problem for Helical Equilibrium States 327 3. General Solutions for Equilibria 330 4. Travelling Waves in Straight Rods 334 5. Buckling under Terminal Thrust and Torque 337 6. Lateral Instability.339 Chapter X. Axisymmetric Equilibria of Cosserat Shells 343 1. Formulation of the Governing Equations 343 2. Buckling of a Transversely Isotropic Circular Plate 348 3. Remarkable Trivial States of Aeolotropic Circular Plates 355 4. Buckling of Aeolotropic Plates 361 5. Buckling of Spherical Shells 364 6. Buckling of Cylindrical Shells 367 7. Asymptotic Shape of Inflated Shells 369 8. Membranes 369 Chapter XI. Tensors 371 1. Tensor Algebra 371 2. Tensor Calculus 378 3. Indicia! Notation 382
xvi CONTENTS Chapter XII. Three-Dimensional Continuum Mechanics 385 1. Kinematics 385 2. Strain 387 3. Compatibility 391 4. Rotation 394 5. Examples 396 6. Mass and Density 400 7. Stress and the Equations of Motion 401 8. Boundary and Initial Conditions 408 9. Impulse-Momentum Laws and the Principle of Virtual Power.. 411 10. Constitutive Equations of Mechanics 417 11. Invariance under Rigid Motions 420 12. Material Constraints 423 13. Isotropy 436 14. Thermomechanics 441 15. The Spatial Formulation 450 Chapter XIII. Elasticity 457 1. Summary of the Governing Equations 457 2. Constitutive Restrictions 459 3. Semi-Inverse Problems of Equilibrium in Cylindrical Coordinates 470 4. Torsion and Other Problems in Cylindrical Coordinates for Incompressible Bodies 474 5. Problems in Cylindrical Coordinates for Compressible Bodies.. 478 6. Deformation of a Compressible Cube into an Annular Wedge and Related Deformations 480 7. Dilatation, Cavitation, Inflation, and Eversion 488 8. Other Semi-Inverse Problems 503 9. Universal and Non-Universal Deformations.; 506 10. Antiplane Problems 509 11. Perturbation Methods 511 12. Radial Motions of an Incompressible Tube 521 13. Universal Motions of Incompressible Bodies 523 14. Standing Shear Waves in an Incompressible Layer 525 15. Linear Elasticity :.. 527 16. Commentary. Other Problems 529 Chapter XIV. General Theories of Rods and Shells 531 1. Introduction. Curvilinear Coordinates 531 2. Rod Theories 535 3. Rods with Two Directors 540 4. Elastic Rods 545 5. Planar Problems for Elastic Rods, Problems with Symmetries.. 548 6. Necking 554
CONTENTS xvii 7. Intrinsic Theories of Rods 558 8. Mielke's Treatment of St. Venant's Principle 559 9. Shell Theories 565 10. Shells with One Director, Special Cosserat Theory, Kirchhoff Theory 568 11. Axisymmetric Deformations of Axisymmetric Shells 572 12. Global Buckled States of a Cosserat Plate 578 13. Intrinsic Theory of Special Cosserat Shells 581 14. Asymptotic Methods. The von Karman Equations 591 15. Commentary. Historical Notes 598 Chapter XV. Nonlinear Plasticity 603 1. Introduction 603 2. Constitutive Equations 604 3. Refinements and Generalizations 609 4. Example: Longitudinal Motion of a Bar 612 5. Antiplane Shearing Motions 617 6. Discrete Models 624 Chapter XVI. Dynamical Problems 629 1. The One-Dimensional Quasilinear Wave Equation 629 2. The Riemann Problem. Uniqueness and Admissibility of Weak Solutions 633 3. Dissipative Mechanisms for Longitudinal Motion. Preclusion of Total Compression 640 4. Shock Structure. Admissibility and Travelling Waves 645 5. Travelling Shear Waves in Viscoelastic Media 649 6. Blowup in Three-Dimensional Hyperelasticity 658 Chapter XVII. Appendix. Topics in Linear Analysis 665 1. Banach Spaces 665 2. Linear Operators and Linear Equations 668 Chapter XVIII. Appendix. Local Nonlinear Analysis 675 1. The Contraction Mapping Principle and the Implicit Function Theorem 675 2. The Lyapunov-Schmidt Method. The Poincare Shooting Method 679 Chapter XIX. Appendix. Degree Theory 683 1. Definition of the Brouwer Degree 683 2. Properties of the Brouwer Degree 687 3. Leray-Schauder Degree 693 4. One-Parameter Global Bifurcation Theorem 697
xviii CONTENTS References 699 Index 737