Abstract
The localized loading of an elastic sheet floating on a liquid bath occurs at scales from a frog sitting on a lily pad to a volcano supported by the Earth’s tectonic plates. The load is supported by a combination of the stresses within the sheet (which may include applied tensions from, for example, surface tension) and the hydrostatic pressure in the liquid. At the same time, the sheet deforms, and may wrinkle, because of the load. We study this problem in terms of the (relatively weak) applied tension and the indentation depth. For small indentation depths, we find that the force–indentation curve is linear with a stiffness that we characterize in terms of the applied tension and bending stiffness of the sheet. At larger indentations, the force–indentation curve becomes nonlinear and the sheet is subject to a wrinkling instability. We study this wrinkling instability close to the buckling threshold and calculate both the number of wrinkles at onset and the indentation depth at onset, comparing our theoretical results with experiments. Finally, we contrast our results with those previously reported for very thin, highly bendable membranes.
Highlights
Poking is a natural way in which to test the material properties of an object, both in everyday life or, more quantitatively, in AFM measurements of graphene [1,2] and biological cells [3]
We have investigated the response of a floating, elastic sheet to an applied, localized load focusing on the limit of low-to-moderate mechanical bendability, 10−2 τ 102
For loads insufficient to wrinkle the sheet, the resultant deformation is axisymmetric and is characterized by two regimes in the force–displacement law: with small displacements, the force is linearly proportional to the imposed indentation, while for large displacements the force is proportional to the square of the imposed indentation
Summary
Poking is a natural way in which to test the material properties of an object, both in everyday life (for example, an under-inflated bicycle tyre) or, more quantitatively, in AFM measurements of graphene [1,2] and biological cells [3]. This corresponds to an object floating on the surface of a liquid: the hydrostatic pressure within the liquid provides a restoring force that is precisely linear in the vertical deflection This linear response is commonly used as a model of an elastic substrate—this model assumes that the substrate consists of an array of linear springs and is known as the mattress model. The material properties of ultra-thin polymer films can be determined from the readily observable wrinkle patterns that form when floating films are subject to a localized force either from the capillary pressure of a fluid droplet or an imposed displacement from an indenter [20,21,22] In both cases, a vertical deflection pulls material radially inwards and in so doing generates a compressive azimuthal stress in the film that results in a radial pattern of wrinkles.
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