Optimal design for rubber concrete layered periodic foundations based on the analytical approximations of band gaps and mapping relations

The seismic performance of rubber concrete-layered periodic foundations are significantly influenced by their design,in which the band gaps play a paramount role.Aiming at providing better designs for these foundations,this study first proposes and validates the analytical formulas to approximate the bounds of the first few band gaps.In addition,the mapping relations linking the frequencies of different band gaps are presented.Furthermore,an optimal design method for these foundations is developed,which is validated through an engineering example.It is demonstrated that ensuring the superstructure’s resonance zones are completely covered by the corresponding periodic foundation’s band gaps can achieve satisfactory vibration attenuation effects,which is a good strategy for the design of rubber concrete layered periodic foundations.

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.

Tunable band gaps in acoustic metamaterials with periodic arrays of resonant shunted piezos

Periodic arrays of resonant shunted piezoelectric patches are employed to control the wave propagation in a two-dimensional （2D） acoustic metamaterial. The performance is characterized by the finite element method. More importantly, we propose an approach to solving the conventional issue of the nonlinear eigenvalue problem, and give a convenient solution to the dispersion properties of 2D metamaterials with periodic arrays of resonant shunts in this article. Based on this modeling method, the dispersion relations of a 2D metamaterial with periodic arrays of resonant shunted piezos are calculated. The results show that the internal resonances of the shunting system split the dispersion curves, thereby forming a locally resonant band gap. However, unlike the conventional locally resonant gap, the vibrations in this locally resonant gap are unable to be completely localized in oscillators consisting of shunting inductors and piezo-patches.

An Analysis of Structural-Acoustic Coupling Band Gaps in a Fluid-Filled Periodic Pipe

A periodic pipe system composed of steel pipes and rubber hoses with the same inner radius is designed based on the theory of phononic crystals. Using the transfer matrix method, the band structure of the periodic pipe is calculated considering the structural-acoustic coupling. The results show that longitudinal vibration band gaps and acoustic band gaps can coexist in the fluid-filled periodic pipe. The formation of the band gap mechanism is further analyzed. The band gaps are validated by the sound transmission loss and vibration-frequency response functions calculated using the finite element method. The effect of the damp on the band gap is analyzed by calculating the complex band structure. The periodic pipe system can be used not only in the field of vibration reduction but also for noise elimination.

Local resonance phononic band gaps in modifiedtwo-dimensional lattice materials

In this paper, modified two-dimensional peri- odic lattice materials with local resonance phononic band gaps are designed and investigated. The design concept is to introduce some auxiliary structures into conventional pe- riodic lattice materials. Elastic wave propagation in this kind of modified two-dimensional lattice materials is studied us- ing a combination of Bloch＇s theorem with finite element method. The calculated frequency band structures of illus- trative modified square lattice materials reveal the existence of frequency band gaps in the low frequency region due to the introduction of the auxiliary structures. The mechanism underlying the occurrence of these frequency band gaps is thoroughly discussed and natural resonances of the auxiliary structures are validated to be the origin. The effect of geo- metric parameters of the auxiliary structures on the width of the local resonance phononic band gaps is explored. Finally, a conceptual broadband vibration-insulating structure based on the modified lattice materials is designed and its capabil- ity is demonstrated. The present work is anticipated to be useful in designing structures which can insulate mechanical vibrations within desired frequency ranges.

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.

Band gaps structure and semi-Dirac point of two-dimensional function photonic crystals

Two-dimensional function photonic crystals, in which the dielectric constants of medium columns are the functions of space coordinates , are proposed and studied numerically. The band gaps structures of the photonic crystals for TE and TM waves are different from the two-dimensional conventional photonic crystals. Some absolute band gaps and semiDirac points are found. When the medium column radius and the function form of the dielectric constant are modulated, the numbers, width, and position of band gaps are changed, and the semi-Dirac point can either occur or disappear. Therefore,the special band gaps structures and semi-Dirac points can be achieved through the modulation on the two-dimensional function photonic crystals. The results will provide a new design method of optical devices based on the two-dimensional function photonic crystals.

Accurate GW_(0) band gaps and their phonon-induced renormalization in solids

Recent years,huge progress of first-principles methods has been witnessed in calculating the quasiparticle band gaps,with many-body perturbation theory in the GW approximation being the standard choice,where G refers to Green’s function and W denotes the dynamically screened Coulomb interaction.Numerically,the completeness of the basis set has been extensively discussed,but in practice far from carefully addressed.Beyond the static description of the nuclei,the electron–phonon interactions(EPIs)are ubiquitous,which cause zero-point renormalization(ZPR)of the band gaps.Therefore,to obtain high quality band gaps,one needs both accurate quasiparticle energies and accurate treatments of EPIs.In this article,we review methods on this.The completeness of the basis set is analyzed in the framework of linearized augmented plane waves,by adding high-energy local orbitals(HLOs).The electron–phonon matrix elements and self-energy are discussed,followed by the temperature dependence of the band gaps in both perturbative and non-perturbative methods.Applications of such an analysis on bulk wurtzite BeO and monolayer honeycomb BeO are given.Adding HLOs widens their GW_(0) band gaps by∼0.4 eV while ZPR narrows them by similar amount.These influences cancel each other,which explains the fortuitous agreement between experiment and theory when the basis set is incomplete and the EPIs are absent.The phonon-induced renormalization,a term often neglected in calculations of the band gaps,is also emphasized by its large magnitude.

Erratum to“Accurate GW_(0) band gaps and their phonon-induced renormalization in solids”

After having Eq.(25),since∑1e-iq.rδvKS(r)/δR1ka|R=R^(0)=∑1e-iq·(r-R1δVKS(r-R1)/δRoka)|R=R^0hasthe lattice periodicity,the matrix inEq.

Comment on “Band gaps structure and semi-Dirac point of two-dimensional function photonic crystals” by Si-Qi Zhang et al.

Recently, Zhang et al. （Chin. Phys. B 26 024208 （2017）） investigated the band gap structures and semi-Dirac point of two-dimensional function photonic crystals, and the equations for the plane wave expansion method were induced to obtain the band structures. That report shows the band diagrams with the effects of function coefficient k and medium column ra under TE and TM waves. The proposed results look correct at first glance, but the authors made some mistakes in their report. Thus, the calculated results in their paper are incorrect. According to our calculations, the errors in their report are corrected, and the correct band structures also are presented in this paper.

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.

A brief review of metamaterials for opening low-frequency band gaps

Metamaterials are an emerging type of man-made material capable of obtaining some extraordinary properties that cannot be realized by naturally occurring materials.Due to tremendous application foregrounds in wave manipulations,metamaterials have gained more and more attraction.Especially,developing research interest of low-frequency vibration attenuation using metamaterials has emerged in the past decades.To better understand the fundamental principle of opening low-frequency(below 100 Hz)band gaps,a general view on the existing literature related to low-frequency band gaps is presented.In this review,some methods for fulfilling low-frequency band gaps are firstly categorized and detailed,and then several strategies for tuning the low-frequency band gaps are summarized.Finally,the potential applications of this type of metamaterial are briefly listed.This review is expected to provide some inspirations for realizing and tuning the low-frequency band gaps by means of summarizing the related literature.

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.

On band gaps of nonlocal acoustic lattice metamaterials:a robust strain gradient model

We have proposed an"exact"strain gradient(SG)continuum model to properly predict the dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions.The key enhancement is proposing a wavelength-dependent Taylor expansion to obtain a satisfactory accuracy when the wavelength gets close to the lattice spacing.Such a wavelength-dependent Taylor expansion is applied to the displacement field of the diatomic lattice,resulting in a novel SG model.For various kinds of diatomic lattices,the dispersion diagrams given by the proposed SG model always agree well with those given by the discrete model throughout the first Brillouin zone,manifesting the robustness of the present model.Based on this SG model,we have conducted the following discussions.(Ⅰ)Both mass and stiffness ratios affect the band gap structures of diatomic lattice metamaterials,which is very helpful for the design of metamaterials.(Ⅱ)The increase in the SG order can enhance the model performance if the modified Taylor expansion is adopted.Without doing so,the higher-order continuum model can suffer from a stronger instability issue and does not necessarily have a better accuracy.The proposed SG continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviors all over the first Brillouin zone.(Ⅲ)The effects of the nonlocal interactions are analyzed.The nonlocal interactions reduce the workable range of the well-known long-wave approximation,causing more local extrema in the dispersive diagrams.The present model can serve as a satisfactory continuum theory when the wavelength gets close to the lattice spacing,i.e.,when the long-wave approximation is no longer valid.For the convenience of band gap designs,we have also provided the design space from which one can easily obtain the proper mass and stiffness ratios corresponding to a requested band gap width.

Syntheses,Structures and Band Gaps of KLnSiS_4(Ln=Sm,Yb)

Two new quaternary sulfides, KSmSiS4 （1） and KYbSiS4 （2）, have been synthesized by high-temperature solid-state reaction. Single,crystal X-ray diffraction analyses indicate that both compounds crystallize in the space group P21/m, and the crystal data are as follows： a = 6.426（11）, b = 6.582（11）, c = 8.602（15）A, β= 107.90（13）°, Z = 2, V= 346.2（10） A^3, Dc = 3.317 g/cm^3, F（000） = 318,μ（MoKα） = 10.334 mm^-1, the final R = 0.0559 and wR = 0.1370 for 1; and α= 6.3244（10）, b = 6.5552（10）, c = 8.5701（15）A, β= 108.001（13）°, Z = 2, V = 337.91（9） A^3, De= 3.621 g/cm^3, F（000） = 334, μ（MoKα） = 15.737 mm^-1, the final R = 0.0422 and wR = 0.0960 for 2. The KLnSiS4 （Ln = Sm, Yb） structure consists of corrugated ∞^2 [LnSiS4]^- layers which are formed by edge-sharing LnS8 bicapped trigonal prisms and SiS4 tetrahedra. The K^＋ cations are located in the cavities defined by S2 anions between the ∞^2[LnSiS4]^- layers. Band-gap analyses show that compounds 1 and 2 are semiconductors with optical band-gaps of 2.40 and 2.34 eV, respectively.

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 0.6046, 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.

Tunable low frequency band gaps and sound transmission loss of a lever-type metamaterial plate

A novel metamaterial plate with subwavelength lever-type resonators is proposed to obtain low frequency broadband band gaps and good sound insulation performance.The band structure is theoretically derived,and the validity of the theoretical method is verified by the finite element method.The formation mechanisms of the band gaps are illustrated by the analysis of the effective dynamic mass density and group velocity.The effect of the lever ratio on the band gaps is analyzed.The results indicate that as the lever ratio increases,the first band gap shifts to lower frequencies,while the bandwidth is widened.Moreover,the sound insulation performance of the proposed metamaterial plate is evaluated via examining the sound transmission loss(STL).Compared with the metamaterial plates without lever accessories,the proposed metamaterial plates with a suitable lever ratio have better sound insulation performance at low frequencies.

Size and temperature effects on band gaps in periodic fluid-filled micropipes

A new model is proposed for determining the band gaps of flexural wave propagation in periodic fluid-filled micropipes with circular and square thin-wall cross-sectional shapes, which incorporates temperature, microstructure, and surface energy effects. The band gaps depend on the thin-wall cross-sectional shape, the microstructure and surface elastic material constants, the pipe wall thickness, the unit cell length, the volume fraction, the fluid velocity in the pipe, the temperature change,and the thermal expansion coefficient. A systematic parametric study is conducted to quantitatively illustrate these factors. The numerical results show that the band gap frequencies of the current non-classical model with both circular and square thin-wall cross-sectional shapes are always higher than those of the classical model. In addition,the band gap size and frequency decrease with the increase of the unit cell length according to all the cases. Moreover, the large band gaps can be obtained by tailoring these factors.

Effects of rotating noncircular scatterers on spin-wave band gaps of two-dimensional magnonic crystals

Using the plane-wave expansion method, the spin-wave band structures of two-dimensional magnonic crystals consisting of square arrays of different shape scatterers are calculated numerically, and the effects of rotating rectangle and hexagon scaterers on the gaps are studied, respectively. The results show that the gaps can be substantially opened and tuned by rotating the scatterers. This approach should be helpful in designing magnonic crystals with desired gaps.

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.