High-frequency long-wave behaviour in elastic waveguides with arbitrary cross-section

Within this proposal we are planning to develop a general methodology to analyze the high-frequency long-wave behaviour of waveguides with arbitrary cross-section. In particular, we formulate a hybrid procedure starting from a long-wave perturbation of the numerical solution of the eigenvalue problems on cross-section for the equations in plane and antiplane elasticity. Two types of cross-sections are studied analytically, including an elliptic and rectangular one. The derived low-dimensional equations will be applied to the analysis of trapped modes in waveguides with shape imperfections. The project is inspired by the theory and practice of non-destructive evaluation and also has potential applications in the optimal design of advanced structures. It generalizes previous considerations of the co-authors and the visiting fellow in the area of longwavelength asymptotics and elastic trapped modes.

Industrial applications of the proposed theoretical research are mainly in the area of non-destructive evaluation, including experimental testing of fatigue cracking in a metal rail which can be modelled as an elastic waveguide with I-shaped cross-section. In addition, a number of simpler systems such as those in optics (bent optical fibres, photonic crystals, fibres embedded within an outer material) also include general cross-section guides. Thus, the mathematical methodology will have important applications. There is also a potential of incorporating the proposed long-wave high-frequency models in commercial specialized software. The proposers are going to communicate with engineers at Brunel University and Imperial College, London, as well as international collaborators from the Universities of Bordeaux and Maine (France), regarding the implementation of the project results. The proposers will explicitly identify the outputs with potential impact in all publications and conference presentations. The areas of potential industrial application will also be highlighted in the materials presented on the Department of Mathematical Sciences research webpage at Brunel University. It will be specified that the proposed theory may be developed for the waveguides with shapes of particular interest for industry: these will include elliptic, rectangular and potentially, I-shaped cross sections. The impact activities will be uniformly distributed between PI, CI and visiting fellow. PI and CI are planning to allocate a PhD student at Brunel University for finite-element modelling of industrially motivated problems. Other academics from the Department of Mathematical Sciences at Brunel University will take part in some of these related activities.