Elasticity and Geometry From hair curls to the non-linear response of shells

We experience elasticity everywhere in daily life: in the straightening or curling of hairs, the irreversible deformations of car bodies after a crash, or the bouncing of elastic balls in ping-pong or soccer. The theory of elasticity is essential to the recent developments of applied and fundamental science, such as the bio-mechanics of DNA filaments and other macro-molecules, and the animation of virtual characters in computer graphics and materials science. In this book, the emphasis is on the elasticity of thin bodies (plates, shells, rods) in connection with geometry. It covers such topics as the mechanics of hairs (curled and straight), the buckling instabilities of stressed plates, including folds and conical points appearing at larger stresses, the geometric rigidity of elastic shells, and the delamination of thin compressed films. It applies general methods of classical analysis, including advanced nonlinear aspects (bifurcation theory, boundary layer analysis), to derive detailed, fully explicit solutions to specific problems. These theoretical concepts are discussed in connection with experiments. Mathematical prerequisites are vector analysis and differential equations. The book can serve as a concrete introduction to nonlinear methods in analysis.

1. Introduction. 2. Three-dimensional elasticity. I: RODS. 3. Equations for elastic rods. 4. Mechanics of the human hair. 5. Rippled leaves, uncoiled springs. II: PLATES. 6. The equations for elastic plates. 7. End effects in plate buckling. 8. Finite amplitude buckling of a strip. 9. Crumpled paper. 10. Fractal buckling near edges. III: SHELLS. 11. Geometric rigidity of surfaces. 12. Shells of revolution. 13. The elastic torus. 14. Spherical shell pushed by a wall. Appendix A: Calculus of variations: a worked example. Appendix B: Boundary and interior layers. Appendix C: The geometry of helices. Appendix D: Derivation of the plate equations by formal expansion from 3D elasticity.

Basile Audoly, CNRS and Université Pierre et Marie Curie, Paris VI, France Yves Pomeau, CNRS, École Normale Supérieure, Paris, France, and University of Arizona, Tucson, USA