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To better understand the appearance of folded structures

What do the convolutions of the brain, the appearance of wrinkles on the skin, formation of mountain ranges and fingerprints have in common? All these structures, yet so different, result from a single process: the compression of a “rigid sheet”. A team involving the Franco-Belgian Statistical Physics Laboratory (CNRS / ENS Paris / Université Paris Diderot / UPMC) and the Laboratory of Interfaces and Complex Fluids at the University of Mons in Belgium has revealed one of the mysteries in the formation of folded structures. These studies provide insight and help predict their occurrence. The results were published on 31 October 2010 in the journal Nature Physics.

Take a thin sheet of solid material and try to compress it so it stays flat. You will not succeed: the sheet bends systematically over its entire length. This is called buckling.

Now stick the sheet on a soft, thick substrate, squeeze it again the same way: this time, it forms very small regular undulations characterized by a certain distance, called period (see image below). You can see these regular folds by squeezing the skin on the back of the hand between the thumb and the forefinger or by leaving a fruit to dry.

If at this stage, you continue to compress the sheet, a totally new and unexpected phenomenon is observed. The folds that were formed separate into two families: one has an increasing amplitude while the other has a decreasing amplitude (see image below). A two-fold concentrates the energy of deformation to create a structure with a period that is twice the initial period. If you increase the compression again, the same process starts again and leads to a quadrupling of the initial period, and so on.

In a particularly original way, the researchers demonstrated that the same mechanism governs the appearance of period doubling in the folded structures and the comings and goings of oscillating systems like a pendulum of variable length (e.g. the censer of the giant cathedral of Saint James of Compostela - the Botafumeiro) or a swing. So there is a similarity between the equations that describe oscillations in space - the case of folded structures - and in time (in the case of pendula of varying lengths).

This example is part of a broader context wherein researchers endeavour to explain the morphogenesis that is induced by mechanical instability. This phenomenon occurs frequently in nature, for example during a folding of geological strata giving rise to mountains or when a living tissue is composed of two layers that develop at different growth rates. The approach used here by scientists can reveal underlying connections between seemingly unrelated phenomena.

The new theoretical model that researchers have developed will make it possible to better understand and therefore predict the occurrence of these folded structures. This work could also have implications in technological fields by opening the door to the development of new methods of micromanufacturing that will shape the material by creating regular micrometric structures.

For any further information, contact Prof. Pascal Damman (UMONS), head of the Laboratory of Interfaces and Complex Fluids at pascal.damman@umons.ac.be