Scientific programme

TC3 "Elastic waves in inhomogeneous periodic structures: filtering effects and localized vibration modes" (20 hrs)

Prof. A.Movchan, Prof. N.Movchan
The course covers the following topics: asymptotic models and numerical illustrations of periodic systems which possess band gaps and support standing waves at low frequencies; structures containing periodic systems of defects, cracks or resonators of different types; tuning mechanisms to control the position of band gaps in dispersion diagrams; dynamics of thick interfaces separating different regions of elastic materials; structural interfaces made up of elastic layers or inertial truss structures; unusual filtering characteristics in the system, which cannot be obtained with multilayered interfaces; applications to metamaterials. The course will consist of 15 hours of lecture time, with 5 hours of tutorials during which the attendees will be given the opportunity to explore further the techniques discussed in the lectures.
The structure of the course is as follows:

• Bloch-Floquet waves in periodic elastic structures: an introduction;
• Bloch waves in spring mass systems, notion of the stop band and group velocity;
• dynamic lattice Green's functions and localised defect modes in infinite discrete systems;
• slowness contours and dynamic anisotropy for higher frequencies;
• hyperbolic metamaterials;
• dynamic fracture in lattice systems;
• transmission of elastic waves through structured interfaces;
• polarisation and filtering of elastic waves.

TC4 "Wiener-Hopf methods with applications to mechanics" (20hrs)

Prof. G.Mishuris, Dr A.Vellender
The course provides fellows with knowledge of advanced analytical methods to solve boundary value problems with applications to solid mechanics. The course includes the following topics: Introduction and historical overview; Fundamentals of complex analysis; Basis of the Wiener-Hopf methods; Applications to solve integral equation on half axis, mixed BVP for crack, problems with imperfect, dynamic problems in solid and structures.
Introduction to the Wiener-Hopf method.

ASYMPTOTICS MINI-COURSE (Dr Vellender):

• Definitions and main ideas
• Regularly perturbed problems (ODEs and algebraic)
• Singularly perturbed algebraic problems
• Applications to simple mechanical systems (including boundary layer analysis)
• Approximation of integrals
• Abelian-type theorems

WIENER-HOPF MINI-COURSE (Prof Mishuris):

• The ideology of Wiener-Hopf techniques
• Solving integral equations
• Solving PDEs with mixed boundary conditions (in a strip and in a half-space)
• Wiener-Hopf methods in a lattice (static and dynamic problems)

We anticipate that the above will be structured as 15 hours of lecture time, with 5 hours of tutorials during which relevant problems will be tackled. Attendees are welcome to bring their own Wiener-Hopf or asymptotic-related problems which may form the basis for discussions.

Invited talk

The scientific programme will be completed with a lecture "Mechanics of thin films on elastic substrates: asymptotic modelling, instabilities, wave propagation, self-similarity" by Prof. Massimiliano Gei (Cardiff University). His talk will be followed by a wine reception in celebration of the 150th anniversary of LMS.

Teachers' profiles

Prof. Alexander Movchan is the Head of the Department of Mathematical Sciences at the University of Liverpool. The areas of his research include perturbation methods, singular integral equations, asymptotic analysis of singularly perturbed boundary value problems, as well as fracture mechanics, Mathematical models of continuum mechanics and of composite media.

Prof. Natasha Movchan is a Professor at of the Department of Mathematical Sciences at the University of Liverpool. Among her research interests are mathematical modelling of fracture in inhomogeneous media and of elastic and acoustic waves interacting with inhomogeneous media; dynamic structural interfaces, spectral problems for elliptic operators and asymptotic analysis of elasticity problems in singularly perturbed domains.

Prof. Gennady Mishuris is a Professor at the Department of Mathematics of Aberystwyth University. He is conducting research into singular integral equations, functional equations of the Wiener-Hopf type, mathematical modelling of problems of biomechanics, and rheology (plastic and visco-plastic materials) and of solid mechanics (such as BVPs with irregular boundaries, singular fields, composites with imperfect interfaces and waves in complex structures). His areas of interest include such engineering applications as crack propagation, hydraulic fracturing, metal forming, material testing and others.

Dr. Adam Vellender received his PhD at Aberystwyth University, where he is now a Research Lecturer. His research interests include mathematical modelling of solids containing cracks and imperfect interfaces; waves in complex structures, singular integral equations and weight function techniques using the Wiener-Hopf method. Dr Vellender has been highly commended for Outstanding Teaching at this year's Student Led Awards, as well as in the Aberystwyth University Exemplary Course Awards.

Invited speaker

Prof. Massimiliano Gei is the Chair of Solid Mechanics and Structures at the School of Engineering of Cardiff University. His research interests include soft smart materials, metamaterials, dynamics of structured media and energy harvesting techniques.