| In recent years, there has been growing interest in the propagation of classical waves in periodic composites, such as electromagnetic waves, elastic waves and acoustic waves. With the in-depth study of photonic crystal, much attention has been focused on the research of the double negative materials, known as metamaterials. Electromagnetic wave will propagate in a unique way in the metamaterial due to its sub-wavelength structure which can exhibit abnormal responses. Analogous to the photonic crystal, the study of the phononic crystal has also made great progress. But metamterials for the elastic waves or acoustic waves have not been proposed. The realization of the double negative metamaterials for elastic wave will provide significant values and potential applications, such as inverse Doppler Effect, negative refraction, sub-wavelength imaging with a flat lens, cloaking and so on.In this thesis, the Multiple Scattering Theory (MST) for three-dimensional elastic systems is derived in detail with the Mie’s scattering matrix of a spherical coated scatter. We also analyze the coupled relationship among three modes. By research on the band gaps of elastic waves, we have more profoundly understood two mechanisms of band gaps, one is from the Bragg diffraction which is sensitive to the periodicity; the other is derived from local resonances of the scatter.By taking the low frequency limitation of the rigorous three-dimensional multiple scattering theory in the fluid matrix, we get the expression of the dynamic effective mass density. It differs from the static version- the volume average of the component mass densities. The physical reason for this difference is explained. This result can be used to predict the velocities of acoustic waves in the low-frequency limitation.Based on the simple analytic model describing the three-component elastic sytems with local resonances, we analyze the band structure and the characteristics in the gap edge. The dipolar resonant drives the core vibrate out of the phase with that of the matrix, which leads to a low frequency resonant gap. The physics origins for the gap clarify the relationship between the resonant frequency and elastic parameters of each component. It gives an intuitive understanding and method to optimize the gap. With Coherent Potential Approximation (CPA) method, we estimate the effective parameters for the three-dimensional composites, including the effective bulk modulus, the effective mass density, and the effective shear modulus, respectively. They are separately connected with Mie’s scattering matrix with different angular scattering channel. It provides a basis for the realization of the double negative metamtareial.The vibration and resonance mechanism indicate that relative movement is the key for the abnormal response. We designed three different resonant units which can provide negative bulk modulus, negative mass density, and negative shear modulus, respectively. The effective elastic parameters will turn to negative when they reach the resonance frequency.We proposed a double unit structure in the three-dimensional stiff solid matrix, with one specially designed to have a strong monopolar resonance while the other one to have a strong dipolar resonance. The former unit composed of bubble-contained water sphere can provide the negative bulk modulus. The latter unit composed of rubber-coated gold spheres can provide the negative mass density. When the two units resonate in the same frequency, double negative metamtaerials for the longitudinal wave will be realized. In addition, the Poisson ratio of this solid based double negative metamterials also becomes negative.Another double unit structure in the two-dimensional stiff solid matrix is also proposed. Two resonant units are composed of the same structure, consisting of rubber-coated gold columns. The monopolar resonance of the unit with thinner coating gives rise to the negative mass density. And the rotation resonance of the unit with thicker coating gives rise to the negative shear modulus. When the two units resonate in the same frequency, a double negative metamtaerial for the transverse wave will be realized in a special direction. |
