| Phononic crystals (PCs) are a kind of artificial functional composite materials with spatial geometric periodicity. One of the distinguishing features of PCs is the presence of band gaps where elastic waves cannot propagate through the systems. In the last two decades PCs have received considerable attention worldwide because of their potential applications in noise reduction, vibration damping, design of new acoustic devices, etc. The study about PCs is focused on analysis of elastic waves propagating behaviors in an explicit way; and the theoretical research through numerical simulations is one of the most used methods. Therefore, it is necessary to develope efficient and convenient numerical methods for the study of PCs. In addition, there are many kinds of interface conditions in various PCs; and different interface conditions may have significant effects on their mechanical behaviors. Thus the interface effect on the wave motion properties of PCs is also an important topic. In this thesis, we will develop a new method based on Dirichlet-to-Neumann (DtN) map, and analyze the effects of various interface conditions on the band structures of PCs. The main contents and results may be summarized as follows:(1) A method based on the cylindrical wave expansions and the DtN map (shortly called as "DtN-map method") is developed with particular attention to the band structures calculation for mixed in-plane wave modes in two-dimensional (2D) PCs of solid/solid, holes/solid and fluid/solid systems. The accuracy, efficiency, convergence, applicability and computational time of this method are analyzed. The results show that the DtN-map method can calculate the band structures of2D PCs with circular cylinder scatterers efficiently with fast convergence and accuracy. The eigenvalue equation which can give the dispersion relations between the eigenfrequency and the Bloch wave vector is linear and involves very small matrix, thus the method has less computational cost, i.e. it is memory-saving and time-saving. The method can yield accurate results for systems with large acoustic mismatch without extra computational cost, thus it is one of the efficient methods for mixed fluid/solid system which is difficult for the plane wave expansion method, the wavelet method, etc.(2) Based on the surface elastic theory, we can calculate the band structures of2D PCs with nano circular holes or inclusions through imposing the Young-Laplce equilibrium equation on the surface/interface, and analyze the surface/interface effects on the band structures. Numerical results show that the surface/interface effects can be characterized by a non-dimensional parameter which is related to the surface modulus, surface residual stress, bulk modulus and hole/inclusion radius. If the parameter is zero, no surface/interface effect is considered. If the parameter is negative (or positive) value, all bands (and thus the band gaps, if exists) become lower (or higher). The surface/interface effect is distinguished when the absolute value of the parameter is bigger.(3) The DtN-map method is developed to calculate the band structures of2D PCs with weakly bonded interface. Both three-phase model and spring-interface model are considered. Comparison of the results of the two models shows that, for out-of-plane wave modes, there are multiple flat bands in the band structures of the three-phase model, but not in the spring-interface model. However, there is a localized resonance bandgap in low frequencies for either the spring-interface model or the three-phase model. With the stiffness of the spring-interface increasing, the localized resonance bandgap rises with its width increases, and it transforms to the Bragg scattering bandgap gradually. That means that the "sofer" the spring-interface is, the stronger the localized resonance is. |
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