Elasticity and geometry : from hair curls to the non-linear response of shells / B. Audoly, Y. Pomeau.

Format:

Book

Author/Creator:

Audoly, B. (Basile)

Contributor:

Pomeau, Yves.

Class of 1924 Book Fund.

Language:

English

Subjects (All):

Elasticity--Mathematics.

Elasticity.

Physical Description:

x, 586 pages : illustrations ; 26 cm

Place of Publication:

Oxford ; New York : Oxford University Press, 2010.

Contents:

1.2 Notations and conventions p. 3

1.3 Mathematical background p. 9

2 Three-dimensional elasticity p. 17

2.2 Strain p. 18

2.3 Stress p. 33

2.4 Hookean elasticity p. 42

2.5 Stress and strain for finite displacements p. 47

Part I Rods

3 Equations for elastic rods p. 59

3.2 Geometry of a deformed rod, Darboux vector p. 61

3.3 Flexion p. 63

3.4 Twist p. 73

3.5 Energy p. 79

3.6 Equilibrium: Kirchhoff equations p. 82

3.7 Inextensibility, validity of Kirchhoff model p. 90

3.8 Mathematical analogy with the spinning top p. 92

3.9 The localized helix: an explicit solution p. 96

4 Mechanics of the human hair p. 105

4.1 Dimensional analysis p. 106

4.2 Equilibrium equations p. 109

4.3 Weak gravity p. 111

4.4 Strong gravity p. 114

4.5 Extensions of the model p. 124

5 Rippled leaves, uncoiled springs p. 132

5.2 Governing equations p. 141

5.3 Helical solutions p. 143

5.4 Godet solutions p. 147

6 The equations for elastic plates p. 157

6.1 Bending versus stretching energy p. 158

6.2 Gauss' Theorema egregium p. 161

6.3 Developable surfaces p. 176

6.4 Membranes: stretching energy p. 184

6.5 Equilibrium: the Föppl-von Kármán equations p. 192

6.6 Elastic energy p. 199

6.7 Narrow plates; consistency with the theory of rods p. 205

7 End effects in plate buckling p. 214

7.1 A historical background on end effects p. 214

7.2 Geometry p. 215

7.3 Governing equations p. 217

7.4 Linear stability analysis p. 221

7.5 Buckling amplitude near threshold p. 224

7.6 Wavenumber selection by end effects p. 231

8 Finite amplitude buckling of a strip p. 251

8.1 A short review on buckling p. 252

8.2 The experiments p. 253

8.3 Equations for the compressed strip p. 256

8.4 Linear stability p. 260

8.5 The Euler column, an exact solution p. 265

8.6 Transition from finite to infinite wavelengths p. 273

8.7 Linear stability of the Euler column p. 282

8.8 Extension of the diagram p. 296

8.9 Comparison with buckling experiments p. 311

8.10 Application: interpretation of delamination patterns p. 314

8.11 Limitations and extensions of the model p. 318

9 Crumpled paper p. 325

9.2 Conical singularities p. 326

9.3 Ridge singularities p. 350

10 Fractal buckling near edges p. 369

10.1 Case of residual stress near a free edge p. 371

10.2 Case of a clamped edge p. 384

Part III Shells

11 Geometric rigidity of surfaces p. 403

11.2 Infinitesimal bendings of a weakly curved surface p. 406

11.3 Infinitesimal bendings: an intrinsic approach p. 408

11.4 Minimal surfaces, Weierstrass transform p. 415

11.5 Surfaces of revolution p. 416

11.6 Crowns: interpretation of rigidity, extension to arbitrary surfaces p. 429

12 Shells of revolution p. 435

12.1 Geometry p. 436

12.2 Constitutive relations p. 442

12.3 Equilibrium of membranes p. 444

12.4 Equilibrium of shells p. 448

12.5 Conclusion, extensions p. 452

13 The elastic torus p. 454

13.2 Mechanical problem p. 457

13.3 Linearized membrane theory p. 460

13.4 Boundary layer equations p. 464

13.5 The curious case of pressurized circular toroidal shells p. 471

13.6 Boundary layer solution for moderate nonlinearity p. 473

13.7 Boundary layer solution for weak nonlinearity p. 475

13.8 Boundary layer solution for strong nonlinearity p. 486

14 Spherical shell pushed by a wall p. 490

14.2 A short account of Hertz' contact theory p. 492

14.3 Point-like force on a spherical shell (Pogorelov) p. 494

14.4 Spherical shell pushed by a plane: overview p. 498

14.5 Equation for spherical shells p. 502

14.6 Spherical shell pushed by a plane: disc-like contact p. 510

14.7 Spherical shell pushed by a plane: circular contact p. 521

14.8 Stability of disc-like contact, transition to circular contact p. 532.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1924 Book Fund.

ISBN:

9780198506256

0198506252

OCLC:

610868433