Patterns formation through elastic instabilities

Hydrodynamics instabilities, such as Benard-Marangoni convection, Rayleigh-Plateau, Rayleigh-Taylor and many others, are well known to produce beautiful patterns (see the book of F. Charru, Hydrodynamics Instabilities). Capillary, viscous and/or inertial forces conspire to generate very regular morphologies. In this seminar, we will see that elasticity can also produce complex structures, either regular or fully random, provided that slender objects are used. These objects are usually classified as rods, shells or sheets. I will focus this talk essentially on constrained sheets. In the "classical picture", thin sheets constrained by external forces minimize elastic energy through focalization of deformation in singularities. We will see that, due to geometric constraints, these origami structures cannot always be obtained. In the second part, we will see that very regular wrinkles can be observed for deformed sheets, if the sheet is "glued" on a soft foundation. At the onset, these wrinkles reflect the competition between various forces. For large deformations, however, the morphology is surprisingly determined by the nature of the foundation. Fold localization or period-doubling bifurcations are indeed observed for liquid or solid substrate, respectively. Finally, these various studies also highlight the effort of this growing interdisciplinary researcher community to the emergence of a global picture of elastic instability.