Neutral Inclusions

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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.