Elastic, acoustic and water wave propagation through inhomogeneous media

Inhomogeneous media (i.e. materials with spatially dependent material properties) are ubiquitous in the world around us. They develop naturally via optimization strategies in our own body (e.g. bone and soft tissues) and the engineering community has made great use of them in an industrial context (e.g. lightweight composite materials used in the aerospace industry designed to be both light but also strong). Frequently, they exhibit complex behaviour which is not characteristic of their individual uniform (homogeneous) constituents. Their increased use in a multitude of applications motivates the need to better understand the way that they behave. Both static and dynamic testing of inhomogeneous materials such as bone and composites takes place in order to better understand their properties but these experiments often prove prohibitively expensive. Analytical modelling can help enormously since firstly it avoids the financial implications of these experiments and secondly it can help to provide insight into the main mechanisms involved in the underlying physics which governs the behaviour of these materials. The fundamental analytical mathematical techniques used in order to better understand inhomogeneous materials are the theories of homogenization, micromechanics and multiple scattering.In this research proposal the principal investigator intends to develop the mathematical theories described above and use them to model a number of diverse problems regarding wave propagation through inhomogeneous media in the fields of solid and fluid mechanics. One example in the area of solid mechanics is the application of the theory to the problem of wave propagation (ultrasound) through bone - this is the principal mechanism by which disease such as osteoporosis can be detected in elderly patients. Mathematical models assist this process by describing the influence of the differing biological and material lengthscales on the effective wavespeed in bone. One of the problems under study will be to assess the effect that microcracks have on wave propagation. In the area of fluid mechanics, one example of a problem to be studied is that of the multiple interaction of (surface) water waves with obstacles and vortices. This interaction has a number of applications such as the effect of waves on off-shore structure stability and also the study of turbulence.This research will be carried out within three research institutes located in Paris, France, where the host researchers are experts in the areas described above.