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 Wiener-Hopf 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.