Active And Passive Controls Of Wave Behaviors In Periodic Structures

Phononic crystals are a kind of artificial periodic functional materials. Similar to the atomic crystals which have the electronic band structure, phononic crystals can block acoustic or elastic waves in specific frequency ranges. This unique property of acoustic band gaps makes phononic crystals a promising application candidate in the fields of noise reduction, acoustic wave control, wave guiding, etc. Nowadays, with the progress of research on phononic crystals, people want to manufacture phononic crystals which not only achieve a certain functionality, but also possess properties that may be flexibly adjusted in service. Accordingly, an active control on phononic crystals which can react to different incident waves is anticipated. In this thesis, both theoretical and numerical methods will be used to deal with periodic structures whose wave behaviors are controlled by either passive or active means. These periodic structures include plates with periodically corrugated surfaces and phononic crystals composed of hyperelastic elastomers. The main contents are listed as follows:(1) For the piezoelectric plates with periodically corrugated surfaces, the Floquet theorem is used in the expansion of the plate waves, and the band structure is calculated. The plate symmetry is found to bear significant influence on the band structure and band gap width. The symmetry breaking of the plate structure, either induced by the corrugation phase difference or the corrugation amplitude difference, will induce more band gaps. The symmetry breaking of the electric boundary conditions will also induce band gaps, which provides us an approach to flexibly control band gaps through electric methods. We also investigated the band structures of wave guides attached to a substrate. The interface between the wave guide and substrate, and the thickness of the substrate will be shown to significantly influence the band structure and band gap width.(2) The supercell plane wave method is used to analyze the band structures of symmetric and antisymmetric plates. The effect of plate symmetry on the defect wave mode in periodically corrugated piezoelectric plates is investigated. We focus on the formation and transportation of the defect bands and find that the defect bands will move down from the upper edge toward the lower edge of the band gap with the increase of the defect length. This phenomenon can be used to tune the defect band and to realize the wave filtering in different frequency ranges. Moreover, the Bragg band gaps are generally not so flat as the non-Bragg ones induced by the mode coupling effect. Hence, the non-Bragg band gaps show a better localization characteristic. Further investigation shows that the symmetry breaking of the plate structure will significantly alter the band structure and defect mode symmetry.(3) The finite deformation theory developed by Ogden and Dorfmann is adopted to obtain the incremental fields superposed on a finitely deformed phononic crystal. By applying the transfer matrix method along with the Bloch theorem, we obtain the band structure of both longitudinal and transverse waves in one-dimensional phononic crystals made of hyperelastic elastomer. Results show that the band gap width is closely related to the effective acoustic impedance difference (EAID) between two materials. The larger the EAID, the wider the band gap. By defining this parameter, one can directly access the information about the band gap width and consequently design the phononic crystals which can be largely deformed under biasing fields more effectively.(4) The finite deformation theory is used to solve the deformation field of elastomeric scatter devices under biasing mechanical load. The incremental wave field superposed on the deformed scatter devices is obtained. The Dirichlet-to-Neumann map method is adopted to obtain the band structure and equi-frequency contours. We observe that different acoustic switches can be realized through the pre-stretches. In the high-frequency range, for instance, the switch between total reflection and negative refraction can be realized. In the middle frequency range, the switch between positive and negative refractions can be realized. And in the lower frequency range, the switch between regular transmission and beam focusing can be realized. These switches will have potential value in wave guiding, focusing images, shielding and receiving acoustic signals, and can be adjusted actively by simple mechanical pre-stretches.