Neutral Inclusions

该项目解决了如何最好地设计复杂材料中的夹杂物的基本问题，以便随意修改/定制局部应力和应变场。夹杂物通常被引入到另一种介质中，以改变整体材料的性能，这些性能可以是热、电、磁、机械等。然而，一旦引入夹杂物，就会导致弱点或应力集中，最终导致材料失效。因此，该项目的挑战是了解夹杂物需要具有的特性，以减少材料失效的影响。这是在线性和非线性弹性状态下。前者已经被研究了一些，但后者是完全未探索。这个问题也与低频透明/隐身领域有很强的联系，在低频透明/隐身领域，通过为物体分配涂层或斗篷来减少物体的散射。波也可以通过适当选择涂层以特定方式操纵。将采取的方法主要是分析，使用偏微分方程来描述手头的物理学，然后使用特定的技术，如渐近和均匀化，以了解夹杂物对周围的场以及介质的整体性质的影响。非线性弹性力学本身就是一个领域，与这种形式的问题相关的理论是不平凡的。虽然有些问题可以通过分析解决，但其他问题则需要计算解决。在非线性弹性领域，学生将寻求允许中性的特定材料行为（应变能函数）。这是一个位于应用数学核心领域的项目，主要是连续介质力学，更具体地说是固体力学，弹性，声学，波和超材料。然而，它与材料科学，特别是复合材料有着密切的联系。

This project addresses the fundamental issue of how best to design inclusions in complex materials such that the local stress and strain fields are modified/tailored at will. Inclusions are typically introduced into another medium to modify the overall material properties, which could be thermal, electrical, magnetic, mechanical, etc. Once introduced however the inclusions can lead to weaknesses or stress concentrations, eventually leading to material failure. The challenge of this project is therefore to understand the properties that inclusions need to have in order to reduce the impact of material failure. This is in both the linear and non-linear elastic regime. The former has been studied somewhat but the latter is completely un-explored. This problem also has strong ties to the area of low-frequency transparency/cloaking, where the scattering from objects is reduced by assigning that object a coating or cloak. Waves can be also be manipulated in specific ways by appropriate choice of coating. The approach that will be taken is predominantly analytical, using partial differential equations to describe the physics at hand and then using specific techniques such as asymptotics and homogenization to understand the effect of the inclusions on the fields around them and also on the overall properties of the medium. Nonlinear elasticity is a field in its own right and the theory associated with problems of this form are non-trivial. Whilst some problems are solvable analytically, others will require computational solution. In the nonlinear elasticity regime, the student will seek specific material behaviour (strain energy functions) that permit neutrality.This is a project that sits in the core area of applied mathematics, and predominantly continuum mechanics, and more specifically solid mechanics, elasticity, acoustics, waves and metamaterials. It however links strongly to materials science and specifically to composite materials.