Regimes of wrinkling in an indented floating elastic sheet

A thin, elastic sheet floating on the surface of a liquid bath wrinkles when poked at its centre. We study the onset of wrinkling as well as the evolution of the pattern as indentation progresses far beyond the wrinkling threshold. We use tension field theory to describe the macroscopic properties of the deformed film and show that the system passes through a host of different regimes, even while the deflections and strains remain small. We show that the effect of the finite size of the sheet ultimately plays a key role in determining the location of the wrinkle pattern, and obtain scaling relations that characterize the number of wrinkles at threshold and its variation as the indentation progresses. Some of our predictions are confirmed by recent experiments of Ripp \emph{et al.} [arxiv: 1804.02421].