Discrete media are a paradigm of metamaterials research that have applications in understanding and designing civil engineering structures, as well as materials with novel waveguiding properties. Often these materials are modelled as being infinite in extent and periodic. There, the corresponding modal analysis can provide a good indication of the structure’s overall behaviour, especially when the medium of interest is large. However, there exists many scenarios where one needs to determine the response of stratified systems subjected to complex loads or possessing inhomogeneities, non-periodic microstructures and/or having multiple boundaries. This webinar will focus on analytical techniques that help to address such problems and enable one to characterise the in-plane waveguiding and scattering properties of two-dimensional elastic periodic systems.In the first part of the talk, we will discuss Lamb wave propagation in microstructured elastic triangular strips attached to an array of gyroscopes. The latter makes the system non-reciprocal and this allows the medium to support uni-directional Lamb waves when subjected to forcing [1]. The solution to this problem is solved using the discrete Fourier transform and we demonstrate how this solution can be exploited to novel waveguides. Namely, we illustrate how we can create networks of structured strips that can channel waves that propagate from one point in the system and along any predefined controllable path in the system to any other point in the network.In the second part of the talk, we investigate the scattering of waves in a triangular elastic lattice by a penetrable inertial line defect [2]. Through the application of the discrete Fourier transform, this problem can be reduced to the analysis of two scalar Wiener-Hopf equations. From there, essential information about all dynamic modes of the system and their symmetry properties can be extracted. This includes all dynamic regimes where localised modes are supported by the defect when interacting with incoming waves. The solution of the Wiener-Hopf equations is represented as a contour integral that can be used to investigate unusual scattering responses of the defect encountered outside of the low-frequency regime.If time permits, we will then discuss a method that allows one to model vibration in finite or infinite non-periodic discrete flexural systems and that allows for a new dynamic homogenisation method that provides an accurate broadband description of the vibration response for infinite structured flexural waveguides [3].All analytical results presented are accompanied by numerical illustrations that demonstrate their effectiveness.Acknowledgement: MJN gratefully acknowledges the support of the EU H2020 grant MSCA-RISE-2020-101008140-EffectFact. MJN would also like to thank the Isaac Newton Institute for Mathematical Sciences (INI) for their support and hospitality during the programme “Mathematical theory and applications of multiple wave scattering" (MWS), where work on some topics from the talk was undertaken and supported by EPSRC grant no. EP/R014604/1. Additionally, MJN is grateful for the funding received from the Simons Foundation that supported his visit to INI during January-June 2023 and participation in MWS programme.
