Unravelling Wrinkle Formation in a Lubricated Viscoplastic Beam

Wrinkles or creases in the surface of a material are indicative of compression. On Earth, mountain ranges formed due to the plate tectonics exhibit regular spaced folds on the surface; and buckles on the surface of ice sheets are observed due to compression from sea ice. Both examples have a layered structure with contrasting rheological behaviours where compression leads to an instability causing the stiff surface to buckle.

This system is often modelled using the Föppl–von-Kármán plate equations for an infinitely wide elastic sheet, leading to an effective beam equation. However, in many cases where the stiff layer is significantly broken-up, or forced to deform well beyond its yield point, other models may be more relevant, such as a substantially more viscous fluid, or a plastic material. In this talk, we consider an infinitely wide viscoplastic sheet which is compressed while lubricated below by a thin layer of viscous fluid. We use the viscoplastic analogue of the Föppl–von-Kármán elastic model and connects viscous sheet models and classical theories of plastic plates.

The time dependent pattern evolution depends on various factors: the length of the sheet, the stretchability or inextensibility, and the presence of a yield stress. For a viscous sheet, there are strong connections to the mode selection and symmetric/antisymmetric patterns observed for floating elastic sheets. When a yield stress is introduced, additional localisation and straight-sided buckles are observed in plastic dominant regimes.