Topics of projects for prospective research students

Asymptotic analysis of fracture in heterogeneous structures with imperfect interfaces
The project involves asymptotic and numerical analysis of singular integral
equations describing cracks in composite media. The integral equation
formulation is set for the displacement field on the crack faces. The main
attention is given to singular perturbations of the domain boundaries
and to the WienerHopf formulations, which occur in the boundary layer problems.

Mathematical modelling of wave propagation in phononic crystals
The aim of this project is to develop a qualitatively new class of mathematical
models describing phononic band gap structures with inertial structural interfaces.
This will involve a combination of continuum and lattice structures and will cover
both infinite periodic structures and layered structures. It is also planned to
study the effect of disorder and models of defects within periodic structures, and
to generalise our earlier models of filters and polarisers of elastic waves developed
for stacks of layers to the more general cases of cylindrical and spherical layered structures.

Spectral problems related to sizing and location of defects in elastic structures
The project involves the mathematical study of spectral problems of
elasticity for solids with small
defects (such as cavities, inclusions and cracks), and it is based on the asymptotic theory
of singular perturbations of elliptic operators. The purpose of this study is to develop
mathematical techniques to detect, locate and size defects in elastic structures.

Mathematical models of dynamic structural interfaces
The project deals with periodic composite structures with inertial interfaces associated with localised
eigenstates of certain spectral problems. Asymptotic algorithms are to be developed to
estimate the frequencies corresponding to localised eigensolutions and to study propagation of
elastic waves in structures of this type.