Flexural Wave Propagation in Microstructured Media. Perfectly Matched Layers and Elastic Metamaterials.

The work is divided into two main topics. In the first part a formulation for Perfectly Matched Layers is given. Surprisingly, such formulation was absent in the scientific literature. In the second part a new type of periodic plate is proposed. In particular, an analytical model of Perfectly Matched Layers (PMLs) for flexural waves within elongated beam structures is given. The model is based on transformation optics techniques and it is efficient both in time harmonic and transient regimes. A comparison between flexural and longitudinal waves is detailed and it is shown that the bending problem requires special interface conditions. A connection with transformation of eigenfrequencies and eigenmodes is given and the effect of the additional boundary conditions introduced at the border of the Perfectly Matched Layer domain is discussed in detail. Such a model is particularly useful for Finite Element analyses pertaining propagating flexural waves in infinite domain. Then, Perfectly Matched Layers for flexural waves are extended to plate structures. Again, the analytical model is based on transformation optics techniques applied on the biharmonic fourth-order partial differential equation describing flexural vibrations in Kirchhoff-Love plates. It is shown that perfect boundary conditions are not an optimal solution, since they depend on the incident waves. The full analytical form of PMLs and zero reflection conditions at the boundary between homogeneous and PML domains are given. The implementation in a Finite Element (FEM) code is described and an eigenfrequency analysis is given as a possible methodology to check the implementation. A measure of the performance of the PMLs is introduced and the effects of element discretization, boundary conditions, frequency, dimension of the PML, amount of transformation and dissipation are detailed. The model gives excellent results also when the applied load approaches the PML domain. In the last part of the work we propose a new type of platonic crystal. The microstructured plate includes snail resonators with low-frequency resonant vibrations. The special dynamic effect of the resonators are highlighted by a comparative analysis of dispersion properties of homogeneous and perforated plates. Analytical and numerical estimates of classes of standing waves are given and the analysis on a macrocell shows the possibility to obtain localization, wave trapping and edge waves. Applications include transmission amplification within two plates separated by a small ligament. Finally we proposed a design procedure to suppress low-frequency flexural vibrations in an elongated plate implementing a by-pass system re-routing waves within the mechanical system.