Formation mechanism of the low-frequency locally resonant band gap in the two-dimensional ternary phononic crystals

The low-frequency band gap and the corresponding vibration modes in two-dimensional ternary locally resonant phononic crystals are restudied successfully with the lumped-mass method. Compared with the work of C. Goffaux and J. Sánchez-Dehesa （Phys. Rev. B 67 14 4301（2003））, it is shown that there exists an error of about 50% in their calculated results of the band structure, and one band is missing in their results. Moreover, the in-plane modes shown in their paper are improper, which results in the wrong conclusion on the mechanism of the ternary locally resonant phononic crystals. Based on the lumped-mass method and better description of the vibration modes according to the band gaps, the locally resonant mechanism in forming the subfrequency gaps is thoroughly analysed. The rule used to judge whether a resonant mode in the phononic crystals can result in a corresponding subfrequency gap is also verified in this ternary case.

Tuning of band-gap of phononic crystals with initial confining pressure

The mechanism of the shift of the band-gap in phononic crystal （PC） with different initial confining pressures is studied experimentally and numerically. The experimental results and numerical analysis simultaneously indicate that the confining pressure can efficiently tune the location in and the width of the band-gap. The present work provides a basis for tuning the band-gap of phononic crystal in engineering applications.

Band Structure of Three-dimensional Phononic Crystals

运用平面波展开法计算由长方体散射物以面心立方结构排列于基体中形成的三维声子晶体的带结构,研究不同组分材料、散射物的填充率和长方体散射物的高与长之比 R_(HL)对带隙的影响.结果表明,将质量密度和波速大的散射体放在质量密度和波速小的基体中所形成的三维(面心立方)固态声子晶体有利于带隙的产生；散射体的填充率为中间值时带隙最宽；散射体的对称性强烈影响带隙,当 R_(HL)大于等于1时,带隙宽度随 R_(HL)的增加而减小,相反,当 R_(HL)小于1时,带隙宽度随 R_(HL)的增加而增加.

Band gaps of elastic waves in 1-D phononic crystals with imperfect interfaces

Band gaps of elastic waves in 1-D phononic crystals with imperfect interfaces were studied. By using the transfer matrix method （TMM） and the Bloch wave theory in the periodic structure, the dispersion equation was derived for the periodically lami- nated binary system with imperfect interfaces （the traction vector jumps or the displacement vector jumps）. The dispersion equation was solved numerically and wave band gaps were obtained in the Brillouin zone. Band gaps in the case of imperfect interfaces were compared with that in the case of perfect interfaces. The influence of imperfect interfaces on wave band gaps and some interesting phenomena were discussed.

Tuning of band gaps for a two-dimensional piezoelectric phononic crystal with a rectangular lattice

In this paper, the elastic wave propagation in a two-dimensional piezoelectric phononic crystal is studied by considering the mechanic-electric coupling. The generalized eigenvalue equation is obtained by the relation of the mechanic and electric fields as well as the Bloch-Floquet theorem. The band structures of both the in-plane and anti-plane modes are calculated for a rectangular lattice by the planewave expansion method. The effects of the lattice constant ratio and the piezoelectricity with different filling fractions are analyzed. The results show that the largest gap width is not always obtained for a square lattice. In some situations, a rectangular lattice may generate larger gaps. The band gap characteristics are influenced obviously by the piezoelectricity with the larger lattice constant ratios and the filling fractions.

INFLUENCES OF ANISOTROPY ON BAND GAPS OF 2D PHONONIC CRYSTAL

Band gaps of 2D phononic crystal with orthotropic cylindrical fillers embedded in the isotropic host are studied in this paper. Two kinds of periodic structures, namely, the square lattice and the triangle lattice, are considered. For anisotropic phononic crystal, band gaps not only depend on the periodic lattice but also the angle between the symmetry axis of orthotropic material and that of the periodic structure. Rotating these cylindrical fillers makes the angle changing continuously; as a result, pass bands and forbidden bands of the phononic crystal are changed. The plane wave expansion method is used to reduce the band gap problem to an eigenvalue problem. The numerical example is given for YBCO/Epoxy composites. The location and the width of band gaps are estimated for different rotating angles. The influence of anisotropy on band gaps is discussed based on numerical results.

Folding beam-type piezoelectric phononic crystal with low-frequency and broad band gap

A folding beam-type piezoelectric phononic crystal model is proposed to isolate vibration. Two piezoelectric bimorphs are joined by two masses as a folding structure to comprise each unit cell of the piezoelectric phononic crystal. Each bimorph is connected independently by a resistive-inductive resonant shunting circuit. The folding structure extends the propagation path of elastic waves, while its structure size remains quite small. Propagation of coupled extension-flexural elastic waves is studied by the classical laminated beam theory and transfer matrix method. The theoretical model is further verified with the finite element method(FEM). The effects of geometrical and circuit parameters on the band gaps are analyzed. With only 4 unit cells, the folding beam-type piezoelectric phononic crystal generates two Bragg band gaps of 369 Hz to1 687 Hz and 2 127 Hz to 4 000 Hz. In addition, between these two Bragg band gaps, a locally resonant band gap is induced by resonant shunting circuits. Appropriate circuit parameters are used to join these two Bragg band gaps by the locally resonant band gap.Thus, a low-frequency and broad band gap of 369 Hz to 4 000 Hz is obtained.

Band structure characteristics of T-square fractal phononic crystals

The T-square fractal two-dimensional phononic crystal model is presented in this article.A comprehensive study is performed for the Bragg scattering and locally resonant fractal phononic crystal.We find that the band structures of the fractal and non-fractal phononic crystals at the same filling ratio are quite different through using the finite element method.The fractal design has an important impact on the band structures of the two-dimensional phononic crystals.

Band Gaps and Single Scattering of Phononic Crystal

A method is introduced to study the transmission and scattering properties of acoustic waves in two–dimen- sional phononic band gap (PBG) materials. First, it is used to calculate the transmission coefficients of PBG samples. Second, the transmitted power is calculated based on the far field approach. We have also calcu- lated the scattering cross section, the results indicate that phononic band gap appear in frequency regions between two well separated resonance states.

Study on Band Gap Characteristics of Boat-Shaped Phononic Crystal

In this paper, we analyze the two-dimensional Boat-shaped structure based on the finite element method. We calculated its energy band structure and vibration transmission properties and found that the structure has band gaps at both high and low frequencies. Compared with common traditional two- dimensional phononic crystals, the boat-shaped phononic crystal has the advantage of larger bandgap design and modulation parameter space due to their structural complexity. In order to obtain better bandgap characteristics, we studied the influence of four key parameters, such as the rod length and the angle between the rods, on the bandgap. The results show that: for low frequency band gaps, the width of the band gap can be effectively changed by changing the size of the angle between the rods while rod length greatly affects the bandgap position;for high band gaps, the length of rods has a large effect on the band gap position. These laws have guiding significance for the bandgap regulation of boat-shaped phononic crystal.

WAVE PROPAGATION IN TWO-DIMENSIONAL DISORDERED PIEZOELECTRIC PHONONIC CRYSTALS

The wave propagation is studied in two-dimensional disordered piezoelectric phononic crystals using the finite-difference time-domain （FDTD） method. For different cases of disorder, the transmission coefficients are calculated. The influences of disorders on band gaps are investigated. The results show that the disorder in the piezoelectric phononic crystals has more significant influences on the band gap in the low frequency regions than in the high frequency ones. The relation between the width of band gap and the direction of position disorder is also discussed. When the position disorder is along the direction perpendicular to the wave transmission, the piezoelectric phononic crystals have wider band gaps at low frequency regions than the case of position disorder being along the wave transmission direction. It can also be found that the effect of. size disorder on band gaps is analogous to that of location disorder. When the perturbation coefficient is big, it has more pronounced effects on the pass bands in the piezoelectric phononic crystals with both size and location disorders than in the piezoelectric phononic crystals with single disorder. In higher frequency regions the piezoelectric effect reduces the transmission coefficients. But for larger disorder degree, the effects of the piezoelectricity will be reduced.

Comparison of the mechanism of gap formation for tri- and bi-component phononic crystal

Using an exact Mie scattering solution, this paper investigates the mode conversions during the Mie scattering of a single bi- or one-component sphere in unbounded epoxy. Then the formation mechanism of the first complete gap in the corresponding tri- or bi-component phononic crystal is investigated by the multiple-scattering method. It is shown that the heavy density of the scatterer plays an essential role in the Mie resonance and the formation of the gaps for both types of the phononic crystals. For the tri-component phononic crystal, the gap is mainly induced by the Mie resonance of the single scatterer. For the bi-component phononic crystal, the transverse wave （by mode-conversion during the Mie scattering under a longitudinal wave incidence） is modulated by the periodicity and governed by the Bloch theory, which induces the gap cooperatively.

Surface effect on band structure of magneto-elastic phononic crystal nanoplates subject to magnetic and stress loadings

This paper presents a theoretical model for the size-dependent band structure of magneto-elastic phononic crystal(PC)nanoplates according to the Kirchhoff plate theory and Gurtin-Murdoch theory,in which the surface effect and magneto-elastic coupling are considered.By introducing the nonlinear coupling constitutive relation of magnetostrictive materials,Terfenol-D/epoxy PC nanoplates are carried out as an example to investigate the dependence of the band structure on the surface effect,magnetic field,pre-stress,and geometric parameters.The results show that the surface effect has promotive influence on dispersion curves of the band structure,and the band gaps can be improved gradually with the increase in the material intrinsic length.Meanwhile,the band gaps exhibit obvious nonlinear coupling characteristics owing to the competition between the magnetic field and the pre-stress.By considering the surface effect and magneto-elastic coupling,the open and closed points of band gaps are found when the lattice constant to thickness ratio increases.The study may provide a method for flexible tunability of elastic wave propagation in magneto-elastic PC nanoplates and functional design of highperformance nanoplate-based devices.

Flexural vibration band gaps in thin plates with two-dimensional binary locally resonant structures

The complete flexural vibration band gaps are studied in the thin plates with two-dimensional binary locally resonant structures, i.e. the composite plate consisting of soft rubber cylindrical inclusions periodically placed in a host material. Numerical simulations show that the low-frequency gaps of flexural wave exist in the thin plates. The width of the first gap decreases monotonically as the matrix density increases, The frequency response of the finite periodic thin plates is simulated by the finite element method, which provides attenuations of over 20dB in the frequency range of the band gaps. The findings will be significant in the application of phononic crystals.

Study on the vibration band gap and vibration attenuation property of phononic crystals

Phononic crystals (PCs) are functional materials with periodic structures and elas- tic wave (vibration) band gaps, where propagation of vibrations with frequencies within band gaps is forbidden. PCs with finite periods can restrain the propagation of vibrations with frequencies in band gaps and thus has vibration attenuation property. Worldwide, many institutions and researchers are engaged in the re- search of PCs, however, studies on the vibration attenuation property of PCs are still limited. In this paper, we report our study of band gaps and vibration attenua- tion properties of 1) a simplified PC—periodic mass-spring structures, 2) longitu- dinal vibration of one-dimensional (1D-), 2D-, 3D-PCs, and 3) the flexural vibration of 1D- and 2D-PCs. These studies provide a foundation for the applications of PCs in vibration attenuation.

Wavelet-based method for computing elastic band gaps of one-dimensional phononic crystals

A wavelet-based method was developed to compute elastic band gaps of one-dimensional phononic crystals. The wave field was expanded in the wavelet basis and an equivalent eigenvalue problem was derived in a matrix form involving the adaptive computation of integrals of the wavelets. The method was then applied to a binary system. For comparison, the elastic band gaps of the same one-dimensional phononic crystals computed with the wavelet method and the well- known plane wave expansion (PWE) method are both presented in this paper. The numerical results of the two methods are in good agreement while the computation costs of the wavelet method are much lower than that of PWE method. In addition, the adaptability of wavelets makes the method possible for efficient band gap computation of more complex phononic structures.

Tunability of Band Gaps in Two-Dimensional Phononic Crystals with Magnetorheological and Electrorheological Composites

The elastic wave propagation properties of phononic crystals(PnCs)composed of an elastic matrix embedded in magnetorheological and electrorheological elastomers are studied in this paper.The tunable band gaps and transmission spectra of these materials are calculated using the finite element method and supercell technology.The variations in the band gap characteristics with changes in the electric/magnetic fields are given.The numerical results show that the electric and magnetic fields can be used in combination to adjust the band gaps effectively.The start and stop frequencies of the band gap are obviously affected by the electric field,and the band gap width is regulated more significantly by the magnetic field.The widest and highest band gap can be obtained by combined application of the electric and magnetic fields.In addition,the band gaps can be moved to the low-frequency region by drilling holes in the PnC,which can also open or close new band gaps.These results indicate the possibility of multi-physical field regulation and design optimization of the elastic wave properties of intelligent PnCs.

Robustly Tuning Bandgaps in Two-Dimensional Soft Phononic Crystals with Criss-Crossed Elliptical Holes

Tuning band gaps in soft materials by post-buckling deformation is becoming an appealing means to manipulate elastic waves. As one of the most favorable topologies, two- dimensional soft structures with circular holes have been extensively studied. Based on the contrarian thinking, this paper starts from the two-dimensional soft structures with criss-crossed elliptical holes, which is close to the post-buckling configuration of soft structures with circular holes, and then proposes to tune the band gaps through elongating or stretching rather than compressing. Influences of the loading magnitude and loading pattern （i.e., uniaxial and biaxial elongations） on the band gaps are studied via the nonlinear finite element simulations. Effects of the geometric parameters （the major-to-minor half-axis ratio and the porosity of the structure） are also discussed. It is shown that, compared with the traditional circular hole case, the band gaps of the unloaded structure with criss-crossed elliptical holes are much richer, and they could be reversely and continuously tuned by tensile loadings. In particular, the deformation is very robust and is insensitive to small geometric imperfections, which is always necessary for triggering the post-buckling deformations. The present work provides a useful reference to the manipula- tion of elastic waves in periodic structures as well as the design of soft phononic crystals/acoustic devices.

Numerical analysis of acoustic band gaps in two-dimensional periodic materials

Using the plane-wave expansion (PWE)method , the band gaps of the two-dimension phononic crystals composed of square, triangle and honeycomb arrays aluminum cylinders in the air are calculated numerically. The band structures of three lattices were compared and analyzed. It is concluded that the band-gap of honeycomb lattices is located at lower frequency fields, compared with square and triangle lattices. When the filling fraction is between 0.091 and 0.6046, the honeycomb lattices have larger band gaps and gain an advantage over square and triangle lattices. In addition, the gap map is introduced to illustrate the influences of filling fraction on the number, the relative width and the limit frequency of the band-gap.

FLEXURAL VIBRATIONS BAND GAPS IN PERIODIC BEAMS INCLUDING ROTARY INERTIA AND SHEAR DEFORMATION EFFECTS

With the idea of the phononic crystals, the beams with periodic structure are designed. Flexural vibration through such periodic beams composed of two kinds of materials is studied. The emphasis is laid on the effects of rotary inertia and shear deformation. Based on the vibration equation, plane wave expansion method is provided. The acceleration frequency responses of such beams with finite structure are simulated by the finite element method. The frequency ranges of sharp drops in the calculated acceleration frequency response curves are in good agreement with those in the band structures. The findings will be significant in the application of the periodic beams.