On Dynamic Homogenization For Elastic Wave Propagation In Periodically Layered Composites

Studies of the propagation of elastic waves in layered composites have long been of interest to researchers in the fields of geophysics, acoustics and nondestructive evaluation. Common to all of these studies is the investigation of the degrees of interaction among the layers, which manifest themselves in the forms of reflection and transmission agents and give rise to geometric dispersion. These interactions depend, among other factors, upon the mechanical properties, geometric arrangements, number and nature of the interfacial conditions and on the loading conditions. The variability in the mechanical properties of the individual layers ranges from the simple case of an isotropic material to the most general anisotropic one, e.g., the triclinic material. Often, the layer’s properties exhibit other effects such as dissipative, piezoelectric and thermal. These effects will undoubtedly result in further complications in the behavior of the system. In this thesis the anti-plane and in-plane problems of wave propagation with oblique incidence of the wave in a composite under perfect/imperfect contact between the layers and periodic distribution between them are studied. Based on an asymptotic dispersive method for the description of the dynamic processes, the dispersion equations were derived analytically from the average model. The main contributions are as follows:(1) A methodology of dynamic homogenization of the wave propagation in a periodic layered composites with perfect/imperfect contact interface, based in an asymptotic dispersive method for the description of the dynamic processes with assuming a single-frequency dependency of the solution for the wave equations is developed. As a result, no new temporal scale is necessary for consideration. Instead, a regular asymptotic expansion for the eigenfrequencies is obtained from the condition of boundedness for the solution.(2) Application of the methodology developed to anti-plane wave propagation with oblique incidence in a periodically layered composite with a perfect contact interface in two-phase isotropic materials is presented. This homogenization formulation provides very satisfactory results for the dispersive nature of the bi-laminate, and gives a better approximation than the lattice model and its approximation model. The results are graphically illustrated for the propagation of an initial disturbance. The differences between the dynamic homogenization and the classical asymptotic homogenization are discussed. The system is more dispersive when the size of the unit cell is increased. Also, the frequency band structure, as a function of the wave-vector components is calculated. The size effects of the microstructure are captured by the local problems.(3) The problem of in-plane wave propagation with oblique incidence and a perfect contact interface using of the methodology is presented. To evaluate the range of validity of this model, we investigate harmonic wave propagation in composites consisting of alternating layers of two isotropic elastic materials. The local problems and the effective macro properties are calculated. Using the present model, dispersion curves are obtained and are compared with those provided by the exact solution and the micro-inertia model. This homogenization formulation provides satisfactory results for a range of wavelengths. The current approach provides the possibility to calculate the frequency band structure as a function of the wave-vector components, and produce contours of constant frequency in the wave-vector components space.(4) Anti-plane and in-plane wave propagation with oblique incidence and an imperfect contact interface by using the methodology is presented. The results show that the imperfect contact introduces more dispersion in the systems. In addition, as the interface stiffness increases, the dimensionless phase velocities increases and converge to perfect contact results. But different behavior is shown when the interface parameter decreases, simulating damage materials. This behavior can be explained by the dimensionless microstructure functions which are functions of the volume fraction. Finally, the auxiliary local functions corresponding to the first local problems are presented, which could be an index to characterize of the damaged area in composites. This model provides a starting point for the study of damage detection in the composites and properties of the smart composites.