Mechanics of thin elastic sheets

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In the last decade, interest in thin materials such as Graphene has exploded. These materials arise in a wide range of applications, including novel electronic devices, as well as coatings of other materials. However, many aspects of the mechanics of materials that are only one or two atoms thick remain poorly understood; indeed in recent years there has been a good deal of controversy in this general area. For example, recent experiments have suggested that the stretching stiffness of Graphene may increase significantly (compared to the accepted value measured in 2008) if the Graphene is first stretched. At the same time, the polymer science community has begun to explore the properties of very thin polymer films, which may be only one or two polymer molecules thick. Here the controversy instead revolves around whether the material properties of such thin films are quantitatively different for thin, versus bulk, materials.A particular problem of interest in a number of systems is the equilibrium of a liquid droplet sitting on or beneath a thin elastic sheet. For example, it has been suggested that the large pressures in drops covered by a sheet of Graphene may give rise to novel chemistry, while changes to droplet shape are used as a diagnostic tool to measure pre-existing stresses in thin films. However, no theoretical predictions for the pressure within a Graphene "nano-bubble" have been made; similarly, the deviations of droplet shape with pre-existing stresses remains contentious with fundamentally different models used by different experimental groups. We will study this fundamental problem from first principles, developing a mathematical framework within which these two specific problems can be properly understood. A key question that we will address is how the equilibrium contact angle of a droplet changes due to the flexibility of the elastic object? The novelty here is that we will develop a formal asymptotic theory valid for relatively weak surface tension (compared to the stiffness of the sheet); this will allow a precise understanding of when approximate results may safely be used experimentally.A second problem of interest concerns the measurement of the mechanical properties of thin elastic objects. At present, experimental groups use ad hoc 'analytical' formulae. While these formulae are constructed to recover the relevant asymptotic limits appropriately, almost all experiments lie in the intermediate regime where the relative errors introduced by the ad hoc formulae are largest. We will develop strategies that will allow experimentalists to test the appropriateness, or otherwise, of such formulae as well as investigating more robust methodologies. A key development will be an understanding of how the behaviour of such thin sheets change when the response is no longer Hookean (i.e. linearly elastic), allowing us to re-interpret recently published experimental data on the stiffening of Graphene under strain and with induced defects.This project falls within the EPSRC "Continuum Mechanics" research area, within the theme "Engineering".