Topics of projects for prospective research students

• Mathematical models of solids with imperfect interfaces

  This project involves analysis of equations of solid mechanics in multi-phase media separated by thin and soft layers. The effective contact conditions involve discontinuities of displacement (and possibly traction) components. The objective of this work is to develop analytical and numerical models describing the fields around imperfect interfaces. This study will require asymptotic theory for singularly perturbed boundary value problems and analysis of singular integral equations. The range of applications involve models (including dynamics) of thin anisotropic plates, shells and models of delamination cracks on imperfect interfaces.

• Asymptotic analysis of crack-defect interaction in three dimensions

  The project deals with asymptotic models of 3D cracks propagating in elastic domains containing small defects (micro-cracks or small inclusions). A defect is characterized by its dipole tensor. A singular perturbation algorithm is to be developed to describe a deflection of the crack front due to interaction with a small defect (or an array of defects).

• Models of periodic structures with defects.

  This project offers analysis of problems of conductivity, electro-magnetism and elasticity in periodic structures with defects. One of the model formulations includes an array of circular inclusions. The unperturbed structure is supposed to be doubly periodic; in the ``damaged'' structure one or several inclusions have been removed. The objective is to develop an analogue of the Rayleigh method describing this damaged structure. The second group of problems involves lattice structures with defects (dislocations) and requires homogenization and numerical analysis to describe the gradient fields around the defects.

Books to read

• A.B. Movchan & N.V. Movchan

  Mathematical modelling of solids with nonregular boundaries, CRC Press 1995
  

• V.A. Kozlov, V.G. Maz'ya & A.B. Movchan

  Asymptotic analysis of fields in multi-structures, Oxford Research Monographs, Oxford University Press 1999
  

• J.R. Ockendon, S.D. Howison, A.A. Lacey & A.B. Movchan

  Applied partial differential equations, Oxford University Press 1999
  Second edition: Oxford University Press, 2003
  

• A.B. Movchan, N.V. Movchan, C.G. Poulton

  Asymptotic models of fields in dilute and densely packed composites, Imperial College Press 2002
  

• editor: A.B. Movchan

  Asymptotics, singularities and homogenisation in problems of mechanics, Kluwer Academic Publishers Group Kluwer Academic Publishers 2003, ISBN: 1402017804
  
  
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