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A new textbook: Elasticity and Geometry, by Audoly and Pomeau

Zhigang Suo's picture

 From hair curls to the nonlinear response of shellsThe publisher sent me the other day this new book by Audoly and Pomeau.  I haven’t gone through the book carefully, but a quick look has indicated that this is a very special book, well worth a close reading.  The book is beautifully written and well produced.  The authors have captured the recent excitement about thin elastic objects, such as rods, plates, and shells.  While existing books on plates and shells mostly focus on calculating critical loads for instability, this new book describes shapes produced by instability.  Examples include wrinkles in a leaf, ridges in a piece of crumpled paper, and curls of a hair.  The subject is photogenic, and the book contains a large number of delightful illustrations.  The book approaches the subject through physical phenomena, rather than mathematical formalisms.  The authors have intended it as a text for a course at the level of senior undergraduate students or beginning graduate students.         

The renewed fascination with thin elastic objects, with an emphasis on pattern formation, seems to have emerged among mathematicians and physicists.  This fascination is contagious and persistent, both for the utility of the results and for the beauty of the phenomena.  In the Foreword to the book, John Hutchinson wrote:

“For almost two centuries, the primary motivation underlying developments in rods, plates and shells derived from the need to have quantitative methods to analyze man-made structures—buildings, bridges, dams, aircraft, etc.  As a consequence, there was relatively little motivation to understand problems involving dramatic geometry changes such as a rod curling about itself or a hemispherical shell turned inside out.  Such behavior falls well outside the envelope of permissible deformations for conventional structures.  Thus, while non-linear geometry does play an important role in the buckling of conventional structures, the research focus was almost always directed to questions such as the maximum load-carrying capacity of a structure, not, for example, its collapsed shape.  And, usually, the shape at maximum load differs slightly from the initial shape.  In recent decades, new motivations began surfacing in fields such as biophysics and biomechanics where extreme deformation shapes and patterns constitute normal behavior.  Nature is replete with examples, at molecular scale and above.  These modern motivations provide the background for Elasticity and Geometry.”

When I was a graduate student, in late 1980s, I took a course on plates and shells from Bernie Budiansky.  He taught from his notes and research papers.  The subject was out of fashion then, but the course left a lasting impression on me.  When John Hutchinson offered the course on plates and shells last year, I sat in, along with a large number of students and postdocs.  Even Howard Stone, a fluid mechanician, dropped in several times.  Over the years, from time to time I wish I had an opportunity to return to the subject.  How about teaching the subject using this new book?  I’ll request our library to order a copy of the book. 

Elasticity of thin objects is also a recent Theme of the iMechanica Journal Club.


arash_yavari's picture

Dear Zhigang:

Could you post the table of contents?


1 Introduction 1

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


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


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


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

Andrea Sansonetti's picture

Good contents, i think i will buy it.

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