Research On The Manipulation Of Flexural Waves By Topological Edge And Higher-order Corner States In Phononic Crystal Plates

Shells or plates are widely used in the fields of aerospace,transportation and energy,where the vibration and noise reduction or vibration control problems have been widely concerned by scientists and engineers.Inspired by the vibration band-gap property in artificial periodic structures like phononic crystals,the research on introducing the periodic cells in plate structures for new approaches for vibration isolation or wave manipulation has appealed much attention.On the other hand,topologically protected edge states have promising potential applications in wave manipulations because they are robust to structural defects and are immune to back backscatterings.In this context,it is of both theoretical and practical significance to carry out the study of wave manipulation in phononic crystal plates by utilizing both topological edge states and high-order corner states.In this thesis the propagation and refraction of topological edge states,and their couplings with higher-order topological corner state are studied,by taking the flexural wave in phononic crystal plate into consideration.The are listed as follows:（1）A phononic crystal plate model with a spring oscillator attached to a homogeneous thin plate is proposed.The topological band gap is opened by breaking the spatial inversion symmetry,and the analogy of the quantum valley Hall effect is realized in the elastic system.The abnormal refraction of the valley topological edge state propagating from the inside the lattice to the surrounding homogeneous medium is extensively investigated.Since the valley topological edge states are locked at theandpoint in the first Brillouin zone,it can be seen that the plane wave component is only determined by the lattice constant,and is independent of the specific frequency.When the resonant parameters are adjusted to make the working frequency drop to a critical value（so that the wavelength is several times of the lattice constant）,the frequency contour of the homogeneous thin plate also shrinks to a critical value with the decreasing of frequency.In this case,the wave vector in any direction of the thin plate cannot match with the tangential component of the topological edge state at the outlet edge.Because the"conservation of wave vector in the tangential direction"cannot be satisfied,the refracted wave cannot propagate into the surrounding medium,instead it can only exist on the outlet edge in the form of an evanescent wave.Further calculations show that the evanescent wave can couple with both the valley topological edge states.（2）Holes with the same size arranged by triangular lattice are introduced in the homogeneous thin plate.In the band structure of the original unit cell with a single hole,the Dirac cone protected by₆symmetry appears at the corner of the first Brillouin zone.Then,a supercell composed of 3×3 primitive cells above is taken as a new one,and the corresponding first Brillouin zone will shrink to 1/3×1/3 of the original one.According to the band folding principle,the original Dirac cone is folded atΓ,thus forming a double Dirac cone.The translation symmetry of the original triangular lattice is broken by perturbating the radius of each hole,and the splitting of the double Dirac cone is realized to open the topological band gap.Two edge states with opposite group velocities were observed at the interface of the two lattices with topologically distinct band gaps,and only one band can be excited by chiral excitations,so as to realize the unidirectional propagation of the edge state.Furthermore,the transmission ability of the topological waveguide is compared with that of the ordinary waveguide in the case of defects or path bending.It is found that the topological waveguide has stronger robustness.（3）Similar to the idea of point（2）,a double Dirac cone is constructed atΓthrough the band folding mechanism in the phononic plate containing hexagonal lattice spring oscillators.The topological band gap is opened by breaking the translation symmetry of the lattice,and then the pseudospin edge state is formed at the interface between the two topological phases.Due to the different crystals on both sides of the topological interface,the overall structure does not meet the strict₆rotation symmetry,so there is a small band gap in the topological edge state.By properly designing the geometry parameters,the width of the band gap can be adjusted to be large enough,so that there are high-order topological corner states in the band gap of this edge state.Finally,using high-order corner states as isolated resonant states（dark mode）and topological edge states as continuous background states（bright mode）,the two topological interfaces are integrated into the same phononic crystal structure so that they have the possibility of coupling.Asymmetric and ultra-sharp line shape Fano resonance（（?）/Δ1）>14000)is observed in the transmission spectrum of edge states calculated by multiple scattering method.The influence of the perturbations of removing oscillator,random rotations,and random distances on the Fano resonance is also considered.The calculation results show that the Fano resonance peak always exists under various perturbations,and the shift of the resonance peak frequency is less than 1%.