The Princess and the Pea: Mathematical Design of Neutral Inclusions and their Fabrication

安徒生的童话《公主与豌豆》描述了一个女孩（后来成为公主）无法在她的床上睡觉，因为她的床垫下面有一颗非常硬的豌豆。尽管豌豆离最上面的床垫有一段距离，但它的影响是向上延伸的，确保了这个女孩因为它的存在而感到不舒服。豌豆在这里的存在类似于夹杂物，它们在结构、复合材料和超材料中无处不在，并且像豌豆一样，它们的存在扰乱了它们所在的背景或基质介质中的局部应力场和应变场。夹杂物的引入是为了提高材料的性能，如刚度、韧性、导热性或导电性等，但由此产生的应力集中往往会导致裂纹的产生和随后的整体材料失效。这可能会在应用中造成灾难性的后果，包括船只沉没和飞机从空中坠落。一个典型的例子是彗星飞机的方形窗户，这导致了飞机在飞行中的结构故障，窗户本身就是飞机外壳中的“包含物”。已经提出并引入了许多缓解这些问题的机制。这包括对内含物形状的精心设计（参见彗星上的窗户），甚至有整本教科书详细描述了特定结构人工制品周围的应力集中。增强剂或涂层也用于夹杂物，尽管这样做的典型目的是改善基体和夹杂物之间的结合（应力集中仍然相对较高）或仅获得复合材料韧性的适度改善。在这个项目中，我们试图开发一个从未被充分利用的概念：中性包容（NI）。这些是涂层夹杂物，涂层的设计是为了确保NI外部的应力场在加载时不受干扰，就好像NI不存在一样。其概念是，涂层可以在一定范围的负载下“掩盖”夹杂物的存在，因此，例如“中性豌豆”不会干扰周围的床垫，从而产生更舒适的床。因此，原则上，NIs能够重新分配应力，降低应力集中（降低失效可能性）并增强材料设计空间，包括更轻，更坚固的材料。尽管在热学和电学应用方面取得了进展，但弹性方程的矢量性质意味着不可能设计出完整的NIs，即对多种加载状态产生中性的NIs。因此，我们称仅在一种加载状态下为中性的ni为“不完全”，因为这种特性大大降低了它们对先进材料应用的吸引力。然而，我们最近的新结果表明，这一瓶颈是由于以前对各向同性NI涂层的过于简单的限制。放宽这一假设并加入各向异性涂层开辟了许多新的机会，包括多种加载状态下的中性，从而产生完全NIs的潜力。在这个项目中，我们将使用数学技术来设计完整的NIs，然后首次制造它们，以期在先进材料应用中部署，从而设计出更轻、更硬、更耐用的材料。该项目将开辟与结构应用材料特性控制和弹性波操纵相关的超材料研究的新途径。

Hans Christian Andersen's fairytale of the ``Princess and the Pea'' describes how a girl (subsequently a princess) cannot sleep in her bed due to the location of a very hard pea under her numerous mattresses. Despite its distance from the top mattress the influence of the pea extends upwards, ensuring that the girl in question is uncomfortable due to its presence. The presence of the pea here is analogous to that of inclusions, which are ubiquitous in structures, composite materials and metamaterials, and like the pea, their presence disturbs the local stress and strain fields in the background or matrix medium in which they reside. Inclusions are introduced to enhance material properties, e.g. stiffness, toughness, thermal or electrical conductivity, etc. but the resulting stress concentrations can often lead to crack initiation and subsequent global material failure. This can have catastrophic consequences in applications, including ships sinking and planes falling out of the sky. A classic example is the Comet aircraft's square window, which contributed to the structural failure of the plane in flight, the window itself here being the ``inclusion'' in the aircraft skin. Numerous mechanisms to mitigate these problems have been proposed and introduced. These include careful design of the shape of inclusions (cf. windows on the Comet) and indeed whole textbooks have been written detailing stress concentrations around specific structural artefacts. Reinforcements or coatings are also used on inclusions, although typically the aim of this is to improve the bonding between the matrix and inclusion (stress concentrations remain relatively high) or to obtain only modest improvements in composite toughness. In this project we seek to develop a concept that has never been fully exploited: that of the Neutral Inclusion (NI). These are coated inclusions, where the coatings are designed to ensure that stress fields exterior to the NIs are unperturbed upon loading, as if the NI was absent. The concept is that coatings can ``cloak'' the presence of the inclusion under a range of applied loadings, so e.g. a ``neutral pea'' would not perturb the mattresses around it, thus yielding a more comfortable bed. In principle therefore, NIs enable a redistribution of stress, a decrease in stress concentration (reduced failure likelihood) and an enhanced materials design space, including lighter, stronger materials.Although advances have been made in thermal and electrical applications, the vectorial nature of the equations of elasticity means that it has not been possible to design complete NIs, i.e. NIs that yield neutrality to more than one loading state. We thus call NIs that are neutral only for one loading state "incomplete", given that this property significantly reduces their attractiveness for advanced materials applications. Our new, very recent results however, illustrated that this bottleneck was due to the previous overly simplistic restriction of isotropic NI coatings. Relaxing this assumption and incorporating anisotropic coatings opens up a host of new opportunities, including neutrality under multiple loading states, giving rise to the potential for complete NIs.In this project we will use mathematical techniques to design complete NIs and then fabricate them for the first time, with a view to deployment in advanced materials applications, enabling the design of lighter, stiffer and more durable materials. The project will open up new avenues in metamaterials research associated with the control of material properties for structural applications and in the manipulation of elastic waves.