Smectic Description of Wrinkle Patterns*

A thin elastic sheet attached to a soft substrate often develops wrinkle patterns when subject to an external forcing or as a result of geometric incompatibility. Such patterns appear spontaneously in a variety of natural systems, ranging from plant tissues to drying paint and from milk skin to human skin. The interplay between elasticity and substrate response favors, at short length scales, a pattern of straight equally spaced wrinkles. This locally preferred order, akin to a two-dimensional smectic liquid crystal, is often in conflict with a large-scale geometric mismatch or a global constraint. The competition gives rise to a wide variety of complex wrinkle patterns, where different intermediate-scale motifs may dominate at different parameter regimes. Among these motifs one finds smooth distortions of the smectic ground state, proliferation of point defects accompanied by amplitude variations, and sharply defined domains separated by thin domain walls.

In this talk I will show how an effective energy functional for the envelope of a wrinkle pattern emerges from coarse graining the elastic energy. I will point to the smectic-resembling terms in this effective energy functional, their impact and their geometry-induced interaction with the remaining terms that are associated with large-scale distortions. I will discuss how the variety of pattern motifs and transitions can be understood through their analogous liquid crystalline phases and phase transitions. I will demonstrate several possible applications of this new understanding of wrinkles.