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。这些材料的应用范围很广，包括新型电子设备，以及其他材料的涂层。然而，只有一两个原子厚的材料力学的许多方面仍然知之甚少；事实上，近年来在这一领域有很多争议。例如，最近的实验表明，如果石墨烯首先被拉伸，石墨烯的拉伸刚度可能会显著增加（与2008年测量的可接受值相比）。与此同时，聚合物科学界已经开始探索极薄聚合物薄膜的性质，这种薄膜可能只有一两个聚合物分子厚。这里的争论围绕着这种薄膜的材料性质是否在数量上不同于薄材料和块状材料。在许多系统中，我们感兴趣的一个特殊问题是液滴在薄弹性片上或下面的平衡。例如，有人提出，被石墨烯片覆盖的液滴中的大压力可能会产生新的化学反应，而液滴形状的变化被用作测量薄膜中预先存在的应力的诊断工具。然而，目前还没有对石墨烯“纳米气泡”内的压力进行理论预测；同样，液滴形状与预先存在的应力的偏差仍然存在争议，因为不同的实验组使用的模型根本不同。我们将从第一性原理出发研究这个基本问题，建立一个数学框架，在这个框架内，这两个具体问题可以得到正确的理解。我们要解决的一个关键问题是液滴的平衡接触角是如何随着弹性物体的柔韧性而变化的？这里的新颖之处在于，我们将开发一种形式渐近理论，适用于相对较弱的表面张力（与板材的刚度相比）；这将使我们能够精确地了解什么时候近似的结果可以安全地用于实验。第二个令人感兴趣的问题是测量薄的弹性物体的机械性能。目前，实验组使用特别的“分析”公式。虽然这些公式的构造是为了适当地恢复相关的渐近极限，但几乎所有的实验都处于中间状态，其中由特设公式引入的相对误差最大。我们将制定策略，使实验学家能够测试这些公式的适当性，以及研究更可靠的方法。一个关键的发展将是理解当响应不再是Hookean（即线性弹性）时，这种薄片的行为是如何变化的，这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".