Smectic Wrinkles and Rankled Smectics

A thin elastic sheet lying on a soft substrate develops wrinkled patterns when subject to an external forcing or as a result of geometric incompatibility. Thin sheet elasticity and substrate response equip such wrinkles with a global preferred wrinkle spacing length and with resistance to wrinkle curvature. In this talk I will show that these features allow the behavior of these systems at intermediate length scales to be described compactly by the theory of smectic liquid crystals in two dimensions. This analogy allows better understanding of the wrinkling patterns seen in such systems, with which I explain pattern breaking into domains, the properties of domain walls, wrinkle undulation, and topological defects in the wrinkling pattern. Defects are of special interest, in wrinkles and in other smectic systems, by virtue of their important role in phase nucleation, pattern reorientation and dynamics. Understanding these phenomena requires the full topological classification of defects and their combination rules, which is vastly convoluted by the integrability conditions associated with the layer structure. I will present an approach that allows for a compact and clear description of these topological rules, specifically for point and line defects in two and three dimensional smectics. I will further utilize this scheme for explaining complex patterns observed in large charge dislocations in three dimensions, and predict the appearance of observable high-order localized point defects, unique to smectic systems.