Beyond the smectic order: splay and amplitude modulations in wrinkle patterns

Thin solid sheets and shells tend to suppress compressive stresses by developing wrinkles -- elongated periodic undulations around a smooth enveloping shape, whose characteristic "wavelength", λ , vanishes with the solid's thickness. While an energetically-favored spacing imparts a local smectic order to the pattern, the geometric constraints underlying the mere presence of wrinkles are often incompatible with a global smectic order – implying localized defects and other meso-scale deviations from an "ideal" array of parallel, elongated wrinkles.
We elucidate this generic phenomenon through numerical simulations of a prototypical model of geometrically incompatible confinement – a thin sheet attached to a ball of harmonic springs. In addition to localized defects of various types, we find that a prominent motif is meso-scale modulations of the wrinkle amplitude in domains characterized by smectic order. This finding motivates a coarse-grained theory that describes the variation of wrinkle patterns at scales much larger than their waevelngth. In addition to a smectic-like order parameter, whose existence was pointed out previously, we show that such a theory must include other fields, resembling the Ginzburg-Landau theory of superconductors.