Wrinkling of a viscoplastic plate on a viscous substrate

We study the formation of wrinkles in a clamped viscoplastic plate overlying a thin viscous substrate, which is compressed horizontally. When the plate is purely viscous, the compressive force in the plate is constant and is set by the velocity boundary conditions. The wavelength of the emergent wrinkles depends on the thicknesses of the two layers and the ratio of the viscosities. As the domain length is reduced relative to the characteristic wavelength, the spatial profile and growth rate start to depend heavily on the domain length and imposed clamped boundary conditions. Introducing a yield stress to the plate initially increases the compressive force, proportional to the Bingham number, due to the requirement for the plate to be yielded throughout the domain. As the bending moments increase in the plate, the compressive force is relieved and a transition occurs where the plate begins to yield through bending rather than compression. During this transition, the plate is dominated by plugged, unyielded regions leading to localisation and the formation of straight-sided wrinkles.