Point defect states of bending waves in phononic crystal thin plates

The band structures of bending waves propagating in the phononic crystal thin plates with point defect are explored based on the supercell technique combined with the improved plane wave expansion method. The plate is composed of a periodic array of circular crystalline Al
₂
O
₃
cylinders embedded in the epoxy matrix for square lattice. The point defect is introduced by changing one of the cylinders' radii. The results show that the defect modes exist in the first band gap, the frequencies and the numbers of the defect modes are strongly dependent on the defect filling fraction and the filling fraction. All the flat defect bands appearing in the first band gap are nondegenerate. For a given defect filling fraction, there are up to five nondegenerate defect bands emerging inside the first band gap, and the frequencies of defect bands increase with the filling fraction increases. The defect bands disappear at very low or very high filling fractions. For a given filling fraction, three nondegenerate defect bands appearing near the upper edge of the gap move to the middle as the defect filling fraction is reduced, and the edges of the first band gap remain hardly changed. The study of the displacement distributions associated with the three defect modes shows that the flat bands correspond to the special localized frequencies of the waves. For the two lower defect modes, the displacement amplitudes around the defect are much bigger than that at or far away from the defect. However, for the higher defect mode, the displacement amplitude reaches the maximum at the location of the defect and decays rapidly with the distance away from that. Obviously, the bending waves correspond to such three modes are so localized at or near the defect that they can not escape into the surrounding phononic crystals. That is, the point defect behaves like a resonant vacant.