Flat, wrinkled, or ridged: Relaxation of an elastic film on a viscous substrate undergoing continuous compression.

We conduct a finite element computational study of the dynamics of a thin elastic film bonded to a much thicker viscous substrate undergoing compression at a fixed rate. The applied compression tends to continuously increase the strain, and hence the elastic energy, of the film. In contrast to the well-studied case of a soft elastic substrate, a viscous substrate cannot store elastic energy; instead it regulates the kinetics of the various mechanisms that dissipate elastic energy of the film. Sufficiently short films remain flat because shear flow in the liquid near the ends allows rapid relaxation of the strain over the entire film length. In longer films, end-relaxation cannot relax film strain in the mid-section, which therefore buckles. Buckles initially appear as packets of approximately-sinusoidal wrinkles. With increasing strain, these packets transform into tall localized ridges separated by nearly flat regions. In all cases, the buckles cause end-relaxation to become dynamically confined to a narrow region near the ends We construct a state map identifying regions of the parameter space of strain vs film length in which the film remains flat, develops wrinkle packets, or develops localized ridges. The evolution of the film energy during continuous compression shows that ridge localization appears due to a competition between two effects: a well-spaced ridges offer a lower energy state than uniform wrinkles, but wrinkles can develop faster because they require the viscous fluid to move over shorter distances.