Mechanics of thin elastic sheets

在过去的十年里，人们对石墨烯等薄材料的兴趣激增。这些材料出现在广泛的应用中，包括新型电子器件以及其他材料的涂层。然而，只有一两个原子厚度的材料的力学的许多方面仍然知之甚少；事实上，近年来在这一普遍领域存在着很大的争议。例如，最近的实验表明，如果首先拉伸石墨烯，石墨烯的拉伸硬度可能会显著增加(与2008年测量的接受值相比)。与此同时，聚合物科学界已经开始探索超薄聚合物薄膜的特性，这种薄膜可能只有一两个聚合物分子厚。相反，争论的焦点是这种薄膜的材料性质对于薄层材料和块状材料在数量上是否存在差异。在许多系统中，一个特别令人感兴趣的问题是位于弹性薄片上或之下的液滴的平衡。例如，有人提出，一层石墨烯覆盖的液滴中的大压力可能会产生新的化学，而液滴形状的变化则被用作测量薄膜中预先存在的应力的诊断工具。然而，对于石墨烯“纳米气泡”内的压力还没有做出理论上的预测；同样，液滴形状与预先存在的应力的偏差仍然存在争议，不同的实验小组使用了根本不同的模型。我们将从第一原理研究这一基本问题，建立一个数学框架，在这个框架内可以正确地理解这两个具体问题。我们要解决的一个关键问题是，由于弹性物体的柔性，液滴的平衡接触角是如何变化的？这里的新奇之处在于，我们将发展一个对相对较弱的表面张力(与薄板的硬度相比)有效的形式渐近理论；这将允许精确地理解何时可以安全地在实验中使用近似结果。第二个感兴趣的问题涉及到薄弹性物体的机械性能的测量。目前，实验小组使用的是特别的“分析”公式。虽然这些公式是为了适当地恢复相关的渐近极限而构造的，但几乎所有的实验都位于特别公式引入的相对误差最大的中间区域。我们将制定战略，允许实验者测试这些公式的适当性或其他方面，以及调查更稳健的方法。一个关键的发展将是理解当响应不再是Hookean(即线弹性)时，这种薄片的行为如何变化，从而使我们能够重新解释最近发表的关于石墨烯在应变和诱导缺陷下的硬化的实验数据。该项目属于EPSRC“连续介质力学”研究领域，主题为“工程”。

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".