Dimensional Phononic Crystal Band Gap

In this paper, the eigenvalue equations of 2D phononic crystals were deduced from the elastic motion equation of the ideally elastic media by plane wave expansion method, during which the Fourier expansion of the crystal lattice periodic functions and Bloch theorem in solid theory were used. Then computational programs were compiled to solve the eigenvalue equations of 2D phononic crystals so that the energy band structures of some periodic composite systems were gained.Energy band structures of solid composite systems (such as polymethyl methacrylate /rubber system) and liquid composite systems (such as chloroform/ carbon tetrachloride system) were studied to confirm that high contrast of physical parameters between the inclusions and matrix is necessary to obtain complete band gaps for elastic composite systems. Generally, materials with small contrast of acoustic parameters can't generate band gaps, the more the parameters' contrast is, the easier the band gap appears. High-density inclusions in a low-density matrix is easier to obtain large band gaps in a periodic solid system(such as nickel/aluminium system), while low-density inclusions in a high-density matrix is better for liquid systems(such as carbon tetrachloride/mercury system) .The influence of the crystal lattice and geometry of the inclusions on phononic band gaps was studied. For a certain phononic crystal, regularly-triangular lattice is better to obtain wide gaps than that of square lattice or hexagonal lattice, and the gaps in periodic media with cylinder inclusions are wider than that of prism inclusions with a cross section of square or 45?rotated square. In carbon tetrachloride/mercury crystal with regularly triangular lattice (lattice constanta = 4cm ) and cylinder inclusions, the widest band gap ΔΩmax =0.6571 is obtained at the fillingfraction y=0.2089, where the ratio of gap width and middle gap is ΔΩ/Ω = 1.0331, and the corresponding gap frequency is ω = 47.4-156.8kHz.The variation of gap width ΔΩ in liquid systems with the filling fraction / was discussed. For a system of low-density inclusions in a high-density matrix, the gap widthbecomes bigger and then smaller with f. While for a system composed of high-densityinclusions and low-density matrix, the gap width becomes bigger consistently with / until itsmaximum value.The influence of the lattice constant a on gap frequency ω was summarized. For a given phononic crystal with certain constituents , lattice structure and filling fraction, there is arelation: caa = 2ΔC0Ω = k (constant), where Ω represents the normalized frequency, and C0 = (for solid systems) or C0 = (for liquid systems).The effect of the filling fraction on phononic band gaps was obtained based on many energy band figures. With the increase of the filling fraction of inclusions, energy bands begin to split and incomplete gap appears. When energy bands continue to split, the number of incomplete gaps increases until the gaps overlap, therefore, complete band gap is obtained.Comparing the band structures calculated by 441 and 841 plane waves, we find that the result of the eigenvalue differs by less than 0.15% up to the eighth band, which means that 441 plane waves are enough to acquire good convergence and accuracy for 2D phononic band structure computation.