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The propagation of long waves along pre-stressed compressible elastic laminates is considered, focusing on anti-plane shear type waves. Within this thesis we investigate the mechanical response of this type of motion for a class of compressible pre-stressed elastic materials in which the strain energy function depends only on the invariants of the strain tensor. The particular emphasis is on small-amplitude motions superimposed on the equilibrium caused by finite deformations. A number of strain-energy functions, including neo-Hookean, Mooney-Rivlin and Varga material models are examined in numerical analysis. The associated dispersion relations are derived by means of the propagator matrix technique, allowing explicit dispersion relations for the considered multi-layer structures, with perfect bonding assumed on the interfaces. The obtained dispersion relations are investigated numerically for three types of possible boundary conditions, including, free faces, fixed faces, and fixed-free faces. This thesis aims at performing a comprehensive long wave asymptotic analysis for the low- and high-frequency regimes in multi-layered structures. The anti-plane assumption allows simple explicit asymptotic results for phase velocity and frequency in terms of elementary functions of the wave number. The derived long wave low- and high-frequency approximations are shown to be in excellent agreement with the exact solution obtained numerically. First, a single pre-stressed layer is examined, subject to all three types of boundary conditions. Then, the consideration is extended for two- and three-layered structures. The associated dispersion relations are analysed, with the corresponding asymptotic approximations constructed. Finally, a specific type of a symmetric 3-layer laminate is considered for all three types of boundary conditions. In view of the symmetry of the laminate, symmetric and anti-symmetric motions are separated for free and fixed face boundary conditions. At the same time, in the fixed-free case, it is not possible to use symmetry about the mid-plane, hence, the results are more algebraically cumbersome.

The thesis deals with 1D and 2D scalar equations governing the dynamic behaviour of coated elastic structures. Low-frequency vibration of a composite rod, beam, rectangular plate and circular plate are studied. The main focus is on physical effects that occur in composite elastic structures with a thin coating. We start with two auxiliary 1D problems for two-component rods and beams. Then elastic waves localised near the edge of a semi infinite plate reinforced by a strip plate are considered within the framework of the 2D classical Kirchhoff theory for plate bending. The boundary value problem for the strip plate is subject to an asymptotic analysis assuming that a typical wave length is much greater the strip thickness. As a result, effective conditions along the interface, corresponding to a plate reinforced by a beam with a narrow rectangular cross-section, are established. They support an approximate dispersion relation perturbed from that for the homogeneous plate with a free edge. The accuracy of the approximate dispersion relation is tested by comparison with the numerical data obtained from the 'exact' matrix relation for a composite plate. The effect of the problem parameters on the localisation rate is studied. In addition, edge bending waves on a thin isotropic semi infinite plate reinforced by a beam are considered within the framework of the classical plate and beam theories. The boundary conditions at the plate edge incorporate both dynamic bending and twisting of the beam. A dispersion relation is derived along with its long-wave approximation. The effect of the problem parameters on the cut-off frequencies of the wave in question is studied asymptotically. The obtained results are compared with calculations for the case when the reinforcement takes the form of a plate strip. Finally, a circular plate reinforced by a thin annular strip of the same thickness is considered. Asymptotic treatment of a strip circular plate with a free outer edge and its inner edge subject to prescribed de ection and rotation is presented. The effective boundary conditions are derived, and approximate dispersion relation is deduced.