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/定制局部应力场和应变场。夹杂物通常被引入到另一种介质中，以改变材料的整体性能，这些性能可以是热、电、磁、机械等。然而，一旦引入夹杂物，就会导致缺陷或应力集中，最终导致材料失效。因此，该项目的挑战在于了解夹杂物的特性，以减少材料失效的影响。这是在线性和非线性弹性状态下。前者已经得到了一些研究，而后者则完全没有被探索过。这个问题也与低频透明/隐身领域密切相关，通过给物体分配涂层或斗篷来减少物体的散射。通过适当选择涂层，也可以以特定的方式操纵波浪。我们将采用的方法主要是分析性的，使用偏微分方程来描述手头的物理现象，然后使用诸如渐近和均匀化等特定技术来理解内含物对其周围场的影响以及对介质整体特性的影响。非线性弹性本身就是一个领域，与这种形式的问题相关的理论是非平凡的。虽然有些问题可以用分析方法解决，但其他问题则需要计算解决。在非线性弹性体系中，学生将寻求允许中性的特定材料行为（应变能函数）。这个项目位于应用数学的核心领域，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，特别是复合材料密切相关。

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.