Wrinkle to fold transition: influence of the substrate response Fabian Brau, * a Pascal Damman, a Haim Diamant b and Thomas A. Witten c Spatially confined rigid membranes reorganize their morphology in response to imposed constraints. Slight compression of a rigid membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad spatial distribution of energy. For larger compression, the deformation energy is progressively localized in small regions which ultimately develop sharp folds. We review the influence of the substrate on this wrinkle to fold transition by considering two models based on purely viscous and purely elastic foundations. We analyze and contrast the physics and mathematics of both systems. 1 Introduction The great variety of structures and shapes in nature have long been a source of amazement and questioning. 1 Understanding how such patterns emerge spontaneously from a homogeneous environment is a key issue in the context of morphogenesis and pattern formation. 2–4 These structures appear in various con- trasted contexts such as during the formation of large-scale structures in the Universe, 5 during thermal convection in uids, Taylor–Couette ow, solidication fronts, oscillatory chemical reactions, 6 during formation of dunes, sand ripples and beach cusps, 7–9 during collisions of ice oes, 10 during formation of icicles, hot-spring landscapes and columnar jointing 11–14 or in minerals like agates. 15 In biology, it is important to know whether the various genetically encoded forms displayed in living organisms are related to physical processes. In this case, simple laws could be identied revealing some universality between disparate systems. 16–22 One important class of patterns is that which emerges spontaneously upon external or internal mechanical constraints applied to the system. Fruits and vegetables, 23,24 wrinkled and damaged skin, 25–27 organs like lungs 28 and arteries, 29 pollen grains 30 or geological folds 31 are some exam- ples of natural systems developing such structures during global deformation induced by differential growth, a drying process or connement. Understanding the origin of these structures emerging from uniform states and identifying the role of mechanical forces in this process is one important aspect of morphogenesis. Controlling the length-scales characterizing these structures allows the fabrication of versatile patterns 32–36 a Laboratoire Interfaces et Fluides Complexes, CIRMAP, Universit´ e de Mons, 20 Place du Parc, B-7000 Mons, Belgium. E-mail: fabian.brau@umons.ac.be b Raymond & Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel c Department of Physics and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA Dr. Fabian Brau obtained his PhD in theoretical physics at the University of Mons in 2001. During a postdoctoral stay at CWI (Amsterdam) in the group of Ute Ebert, where he studied instabilities of ionization fronts, his interests moved from hadronic physics to nonlinear sciences. He then joined the group of Pascal Damman to work on elastic instabilities. His recent research focuses on pattern formation, thin lms and front instabilities, free moving boundary problems and Laplacian growth. Dr. Pascal Damman obtained his PhD in crystallography of supramolecular assemblies at the University of Mons in 1992. Aer a postdoctoral training at the Institut Charles Sadron in Strasbourg, he was appointed as a research associate of the FNRS. He is now Professor of the Department of Chemistry, head of the Complex System Research Institute of the University of Mons. His research focuses on the stability of uids conned in thin lms and the design of complex structures from mechanically unstable plates. Cite this: Soft Matter, 2013, 9, 8177 Received 5th March 2013 Accepted 31st May 2013 DOI: 10.1039/c3sm50655j www.rsc.org/softmatter This journal is ª The Royal Society of Chemistry 2013 Soft Matter, 2013, 9, 8177–8186 | 8177 Soft Matter REVIEW