Modeling nonlinear problems in the mechanics of strings and rods : the role of the balance laws

목차 (Table of Contents)

• CONTENTS
• Part Ⅰ Mechanics of Strings
• 1 Mechanics of a String = 3
• 1.1 Introduction = 3
• 1.2 Notation and Nomenclature = 4
• 1.3 Space Curves = 5
• 1.3.1 The Frenet Triad, Torsion, and Curvature = 6
• 1.3.2 The Frenet-Serret Relations = 8
• 1.3.3 A Plane Curve = 9
• 1.3.4 A Circular Helix = 10
• 1.4 A Material Curve = 13
• 1.4.1 Stretches, Derivatives, and Velocities = 14
• 1.4.2 Functions and Their Derivatives = 16
• 1.4.3 Discontinuities = 17
• 1.4.4 Eulerian Formulation = 19
• 1.4.5 Superposed Rigid Body Motions = 21
• 1.5 Balance Laws = 22
• 1.5.1 Assigned Forces, Contact Forces, and Material Forces = 23
• 1.5.2 The Postulated Balance Laws = 25
• 1.5.3 Localization Procedure = 26
• 1.5.4 Local Balance Laws = 27
• 1.5.5 Jump Conditions = 28
• 1.6 Elastic Strings and Inextensible Strings = 30
• 1.6.1 Gibbs Free Energy = 33
• 1.6.2 Inextensibility = 34
• 1.6.3 Identities = 34
• 1.7 Summary of the Governing Equations = 35
• 1.8 An Elementary Example Involving Material Forces = 37
• 1.8.1 Interpretations of B1 and C = 39
• 1.8.2 A Uniform Bar = 42
• 1.9 Closing Remarks = 42
• 1.10 Exercises = 43
• 2 Applications of the Mechanics of a String = 49
• 2.1 Introduction = 49
• 2.2 Steady Axial Motions of Elastic Strings and Inextensible Strings... = 51
• 2.2.1 Closed Loops of String = 53
• 2.3 Inextensible Strings with Shocks = 56
• 2.3.1 General Considerations = 57
• 2.3.2 Boundary Conditions, Sources, Sinks, and Reservoirs = 58
• 2.4 Cayley's Problem = 60
• 2.5 A Chain of Finite Length Falling off the Edge of a Table = 64
• 2.6 "Chains that Suck" = 68
• 2.7 The Falling Folded Chain = 72
• 2.8 The Chain Fountain = 78
• 2.8.1 An Inverted Catenary = 78
• 2.8.2 The String Leaving the Heap = 81
• 2.8.3 The String at the End of the Catenary = 82
• 2.8.4 Characteristics of the Chain Fountain = 83
• 2.9 Closing Comments = 86
• 2.10 Exercises = 87
• 3 Link, Writhe, and Twist = 93
• 3.1 Introduction = 93
• 3.2 Space Curves, Ribbons, and Framings = 94
• 3.2.1 Ribbons = 98
• 3.2.2 Gauss-Bonnet Theorem = 99
• 3.3 Gauss' Linking Number of Two Space Curves = 101
• 3.4 Total Geometric Torsion of a Space Curve and Total Twist of a Ribbon = 106
• 3.5 Calugareanu's Theorem = 108
• 3.6 Examples of Computing Writhing Numbers = 111
• 3.7 Self-Linking of a Space Curve with Application to Strands of DNA = 113
• 3.8 Exercises = 115
• Part Ⅱ Mechanics of Rods
• 4 Theory of the Elastica and a Selection of Its Applications = 121
• 4.1 Introduction = 121
• 4.2 Kinematical Considerations = 123
• 4.3 Balance Laws = 127
• 4.3.1 Local Balance Laws and Constitutive Relations = 128
• 4.3.2 Jump Conditions = 130
• 4.3.3 Summary of the Governing Equations = 132
• 4.4 A Terminally Loaded Elastica and the Kinetic Analogue = 132
• 4.5 The Adhesion of a Rod = 137
• 4.5.1 General Considerations = 138
• 4.5.2 Summary of the Solution Procedure = 140
• 4.5.3 Examples = 143
• 4.6 The Elastica Arm Scale = 147
• 4.6.1 Background = 148
• 4.6.2 The Deformable Arm Scale = 149
• 4.6.3 The Operation of the Arm Scale = 153
• 4.6.4 Insights from a Pair of Pendula = 154
• 4.7 Potential Energies and a Variational Formulation = 161
• 4.7.1 A Terminally Loaded Rod Deforming Under a Conservative Assigned Force = 161
• 4.7.2 Application to an Adhesion Problem = 164
• 4.8 Conditions for Stability from the Second Variation = 167
• 4.8.1 A Representation for the Second Variation = 168
• 4.8.2 Conjugate Points and the Riccati and Jacobi Equations = 171
• 4.8.3 The Criterion N1 = 172
• 4.8.4 The Criterion B1 = 173
• 4.8.5 The Criterion S1 = 173
• 4.9 Simple Examples of Buckling = 174
• 4.9.1 Compressing an Adhered Rod = 174
• 4.9.2 Buckling of a Clamped Rod = 176
• 4.9.3 Stability of Peeling = 178
• 4.10 Additional Areas of Application of the Elastica = 182
• 4.11 Exercises = 183
• 5 Kirchhoff's Rod Theory = 187
• 5.1 Introduction = 187
• 5.2 The Directed Curve = 189
• 5.2.1 An Approximation = 189
• 5.3 Kinematics of Kirchhoff's Rod Theory = 191
• 5.3.1 Parameterizations of the Rotation Tensor = 193
• 5.3.2 Strains = 197
• 5.3.3 Inertias = 200
• 5.4 Further Kinematics and Discontinuities = 202
• 5.5 Momenta and Kinetic Energy = 204
• 5.6 A Strain Energy Function = 205
• 5.7 Balance Laws for the Rod Theory = 207
• 5.7.1 Assigned Forces, Assigned Director Forces, and Assigned Moments = 208
• 5.7.2 Conservation Laws = 210
• 5.8 Local Balance Laws and Jump Conditions = 211
• 5.8.1 Local Balance Laws = 211
• 5.8.2 Jump Conditions = 212
• 5.9 Constitutive Relations = 213
• 5.10 Twist, Torsion, and Tortuosity = 215
• 5.11 Summary of the Governing Equations for Kirchhoff's Rod Theory = 217
• 5.12 Relation to the Theory of the Elastica = 218
• 5.13 A Terminally Loaded Rod and a Kinetic Analogue = 219
• 5.13.1 Kirchhoff's Kinetic Analogue = 220
• 5.14 The Simplest Problem : Bending and Torquing into a Helical Shape = 222
• 5.15 Hockling of Cables, Loop Formation, and Localized Buckling = 225
• 5.15.1 Development of the Reduced Dynamical System = 227
• 5.15.2 The Straight Rod and a Pair of Rods Bent into a Helical Form = 229
• 5.15.3 Nontrivial and Localized Equilibrium States of the Rod = 233
• 5.15.4 Comments = 239
• 5.16 Rods with Intrinsic Curvature : Tendril Perversion, Helical Solutions, and Buckling = 240
• 5.16.1 Balance Laws for an Intrinsically Curved Rod = 241
• 5.16.2 Helical Solutions : Solving for the Geometric Torsion, Curvature, and Angle of Twist = 241
• 5.17 Buckling of a Clamped Rod = 247
• 5.18 Spiral Springs and a Stiffness Matrix = 252
• 5.18.1 The Reference Configuration = 253
• 5.18.2 The Balance Laws = 254
• 5.18.3 A Stiffness Matrix = 255
• 5.19 Closing Remarks = 258
• 5.20 Exercises = 258
• 6 Theory of an Elastic Rod with Extension and Shear = 269
• 6.1 Introduction = 269
• 6.2 Kinematical Considerations = 270
• 6.3 Summary of the Governing Equations for the Rod Theory = 273
• 6.3.1 Constitutive Relations for n and m = 274
• 6.4 Treatments of Material Symmetry = 277
• 6.4.1 The Case of a Constant Transformation Q = 278
• 6.4.2 Transverse Isotropy and Transverse Hemitropy = 281
• 6.4.3 Application to Kirchhoff's Rod Theory = 284
• 6.5 Application to Torsion and Extension = 284
• 6.6 Ericksen's Uniform States = 287
• 6.6.1 Kinematical Considerations = 288
• 6.6.2 Forces and Moments = 290
• 6.7 Closing Comments = 290
• 6.8 Exercises = 291
• 7 Green and Naghdi's Rod Theory = 295
• 7.1 A Hierarchy of Rod Theories = 295
• 7.2 Kinematics of Green and Naghdi's Rod Theory = 296
• 7.2.1 Three-Dimensional Considerations = 298
• 7.3 Strains = 298
• 7.3.1 Infinitesimal Strains = 299
• 7.4 Momenta and Kinetic Energy = 302
• 7.5 The Strain Energy Function = 303
• 7.5.1 Infinitesimal Theory = 305
• 7.6 Conservation Laws for the Rod Theory = 307
• 7.6.1 Assigned Forces and Director Forces = 307
• 7.6.2 Singular Supplies = 309
• 7.6.3 Conservation Laws = 311
• 7.7 Local Balance Laws and Jump Conditions = 312
• 7.7.1 Local Balance Laws = 312
• 7.7.2 Jump Conditions = 314
• 7.8 Constitutive Relations = 315
• 7.8.1 Constrained Elastic Rods = 316
• 7.9 Summary of the Governing Equations for an Elastic Rod = 318
• 7.10 Boundary Conditions = 319
• 7.11 Kirchhoff's Rod Theory as a Constrained Rod Theory = 320
• 7.12 Prescriptions and Three-Dimensional Considerations = 323
• 7.12.1 Kinematical Considerations = 323
• 7.12.2 Inertias = 325
• 7.12.3 Preliminary Identities Pertaining to Volume Integrals = 325
• 7.12.4 Integrating the Balance of Linear Momentum = 328
• 7.12.5 Balance of Linear Momentum for the Directed Curve = 329
• 7.12.6 Balances of Director Momenta for the Directed Curve = 330
• 7.12.7 The Moment of Momentum and the Energy Balances for the Directed Curve = 331
• 7.12.8 Balance of Material Momentum for the Directed Curve = 332
• 7.12.9 Comments on the Integration Process = 334
• 7.13 Applications = 335
• 7.14 Exercises = 335
• Part Ⅲ Background Material
• 8 A Rapid Review of Some Elements of Continuum Mechanics = 345
• 8.1 Introduction = 345
• 8.2 Some Kinematical Results = 346
• 8.2.1 Curvilinear Coordinates = 347
• 8.2.2 A Material Curve = 350
• 8.2.3 Metric Tensors and Identities = 351
• 8.2.4 Convected Coordinates = 351
• 8.3 Stress Tensors and Divergences = 355
• 8.3.1 Divergences = 356
• 8.3.2 The Traction Vector and a Divergence = 357
• 8.4 Balance Laws = 359
• 8.5 Invariance Requirements under Superposed Rigid Body Motions = 359
• 8.6 Constitutive Relations for Hyperelastic Bodies = 361
• 8.6.1 A Mooney-Rivlin Material = 364
• 8.6.2 Additional Remarks = 365
• 8.7 Configurational, Material, or Eshelbian Forces = 365
• 8.8 A Material Momentum Balance Law = 366
• 8.8.1 The Local Form = 367
• 8.8.2 The Jump Condition = 369
• 8.9 Closing Comments = 370
• 8.10 Exercises = 371
• 9 Variational Methods = 377
• 9.1 Introduction = 377
• 9.2 Variations and Necessary Conditions = 378
• 9.3 The Euler-Lagrange Necessary Condition = 379
• 9.3.1 Varying Boundary Conditions = 381
• 9.3.2 Weierstrass-Erdmann Corner Conditions = 382
• 9.3.3 Examples = 384
• 9.4 Hamilton's Principle of Least Action = 387
• 9.5 The Wave Equation = 387
• 9.6 Application to Green and Naghdi's Rod Theory = 389
• 9.7 Closing Remarks = 390
• References = 393
• Index = 423