WAVE PROPAGATION IN ELASTIC MEDIA WITH INTERNAL STRUCTURE. PERIODIC TRANSFORMATIONS AND CURVED BEAMS

In this thesis we will focus on the linear elastodynamic properties of complex materials. In Chapter 2, we review the linear theory of elasticity. This chapter provides a brief overview of the basic laws of elasticity theory for di_erent coordinate systems, that will facilitate the further development of our research. Dispersion properties of 2D periodic systems for di_erent lattices are reported in Chapter 3. The governing equations of out of plane wave and examples of coordinate transformations in Cartesian and cylindrical are considered in Chapter 4. Also the scattering problems by square and cylindrical uncloaked and cloaked holes are investigated analytically and numerically. In Chapter 5 a periodic transformation approach has been applied to the problem of out of plane shear wave propagation in an isotropic linear elastic material. The Chapter is organized as follows. In Section 5.1 we present initial and transformed equations of motion and corresponding boundary conditions, describing the periodic locally radial geometric transformation. In Section 5.2 we report the comparative analysis of dispersion properties and briey describe the applied multipole expansion method. In particular, we focus our attention on classical, overlapping and unfolding transformations by also performing a low-frequency, long wavelength homogenisation. In Section 5.3 we show several application including a transmission problems in a continuum and in a waveguide, the detection of defect modes and the design of the transformation for the existence of Dirac points. In Chapter 6, the mathematical model of a curved beam that is connected to two semi-in_nite straight beams is developed. Dispersion properties of curved beams are derived, characterized by three di_erent propagating regimes. By implementing the Transfer matrix approach, the reection and transmission coe_cients that depend on the curvature, frequency and total angle of the curved beam are determined. By analysing the e_ect of the curvature, frequency and total angle on energy ux, separation between high frequency/low curvature regime, where the incident wave is practically totally transmitted, and low frequency/high curvature regime where, in addition to reection there is a strong coupling between longitudinal and exural waves, are de_ned. Finally, general conclusions are given in the last chapter.