B. Audoly and Y. Pomeau
Elasticity and Geometry
Oxford University Press (2010)
1 Introduction1
1.1 Outline 1
1.2 Notations and conventions 3
1.3 Mathematical background 10
2 Three-dimensional elasticity 18
2.1 Introduction 18
2.2 Strain 19
2.3 Stress 35
2.4 Hookean elasticity 45
2.5 Stress and strain for finite displacements 50
2.6 Conclusion 59
PART I RODS
3 Equations for elastic rods 65
3.1 Introduction 65
3.2 Geometry of a deformed rod, Darboux vector 67
3.3 Flexion 69
3.4 Twist 80
3.5 Energy 86
3.6 Equilibrium: Kirchhoff equations 89
3.7 Inextensibility, validity of Kirchhoff model 98
3.8 Mathematical analogy with the spinning top 100
3.9 The localized helix: an explicit solution 104
3.10 Conclusion 109
4 Mechanics of the human hair 113
4.1 Dimensional analysis 114
4.2 Equilibrium equations 117
4.3 Weak gravity 120
4.4 Strong gravity 122
4.5 Extensions of the model 133
4.6 Conclusion 137
5 Rippled leaves, uncoiled springs 141
5.1 Introduction 141
5.2 Governing equations 150
5.3 Helical solutions 153
5.4 Godet solutions 157
5.5 Conclusion 163
PART II PLATES
6 The equations for elastic plates 169
6.1 Bending versus stretching energy 170
6.2 Gauss’ Theorema egregium 173
6.3 Developable surfaces 188
6.4 Membranes: stretching energy 196
6.5 Equilibrium: the F¨oppl–von K´arm´an equations 205
6.6 Elastic energy 212
6.7 Narrow plates: consistency with the theory of rods 218
6.8 Discussion 223
6.9 Conclusion 226
7 End effects in plate buckling 228
7.1 A historical background on end effects 228
7.2 Geometry 229
7.3 Governing equations 231
7.4 Linear stability analysis 236
7.5 Buckling amplitude near threshold 239
7.6 Wavenumber selection by end effects 246
7.7 Experiments 263
7.8 Conclusion 265
8 Finite amplitude buckling of a strip 267
8.1 A short review on buckling 268
8.2 The experiments 269
8.3 Equations for the compressed strip 272
8.4 Linear stability 276
8.5 The Euler column, an exact solution 282
8.6 Transition from finite to infinite wavelengths 289
8.7 Linear stability of the Euler column 299
8.8 Extension of the diagram 314
8.9 Comparison with buckling experiments 329
8.10 Application: interpretation of delamination patterns 332
8.11 Limitations and extensions of the model 336
8.12 Conclusion 339
9 Crumpled paper 344
9.1 Introduction 344
9.2 Conical singularities 345
9.3 Ridge singularities 370
9.4 Conclusion 387
10 Fractal buckling near edges 390
10.1 Case of residual stress near a free edge 392
10.2 Case of a clamped edge 406
10.3 Summary and conclusion 420
PART III SHELLS
11 Geometric rigidity of surfaces 425
11.1 Introduction 426
11.2 Infinitesimal bendings of a weakly curved surface 428
11.3 Infinitesimal bendings: an intrinsic approach 430
11.4 Minimal surfaces, Weierstrass transform 437
11.5 Surfaces of revolution 438
11.6 Crowns: interpretation of rigidity, extension to arbitrary surfaces 452
11.7 Conclusion 455
12 Shells of revolution 458
12.1 Geometry 459
12.2 Constitutive relations 465
12.3 Equilibrium of membranes 467
12.4 Equilibrium of shells 471
12.5 Conclusion, extensions 475
13 The elastic torus 478
13.1 Introduction 478
13.2 Mechanical problem 481
13.3 Linearized membrane theory 484
13.4 Boundary layer equations 489
13.5 The curious case of pressurized circular toroidal shells 496
13.6 Boundary layer solution for moderate nonlinearity 497
13.7 Boundary layer solution for weak nonlinearity 500
13.8 Boundary layer solution for strong nonlinearity 511
13.9 Conclusion 513
14 Spherical shell pushed by a wall 516
14.1 Introduction 516
14.2 A short account of Hertz’ contact theory 518
14.3 Point-like force on a spherical shell (Pogorelov) 520
14.4 Spherical shell pushed by a plane: overview 524
14.5 Equation for spherical shells 529
14.6 Spherical shell pushed by a plane: disc-like contact 537
14.7 Spherical shell pushed by a plane: circular contact 548
14.8 Stability of disk-like contact, transition to circularcontact 559
Appendix A Calculus of variations: a worked example 574
A.1 Model problem: the Elastica 574
A.2 Discretization of energy using a Riemann sum 576
A.3 Calculus of variations: the Euler-Lagrange method 577
A.4 Handling additional constraints 579
A.5 Linear stability analysis 580
A.6 Exact solution 581
Appendix B Boundary and interior layers 586
B.1 Layer at an interior point 586
B.2 Layers near a boundary 594 References 596
Appendix C The geometry of helices 597
Appendix D Derivation of the plate equations by formal expansion from 3D elasticity 599
D.1 Introduction 599
D.2 Scaling assumptions 601
D.3 Basic equations 602
D.4 Expansion of the basic quantities 603
D.5 Solution at leading order 604
D.6 Conclusion 609
Index 611