Band Structures Of Elastic Waves Research Articles

Elastic foundation-introduced defective phononic crystals for tunable energy harvesting

Defective phononic crystals offer the advantage of concentrating elastic waves, thereby enhancing the potential for piezoelectric energy harvesting (PEH). However, a key limitation is their reliance on a fixed operating frequency, rendering them susceptible to the prevailing vibration environment. To surmount this constraint, this study introduces a novel approach involving a tunable elastic foundation system for defective phononic crystal structures. The newly developed phononic crystal is fashioned by integrating a periodically elastic foundation beneath a uniform beam. A defect is induced by selectively removing specific elastic foundations and integrating piezoelectric components. Explicit analytical solutions are established through the transfer matrix method and the spectral element method, which are subsequently corroborated via comparison with finite element results. The findings underscore that the periodic elastic foundations impart bandgaps in the elastic wave band structure. The absence of specific elastic foundations results in the emergence of distinct defect modes. Additionally, frequency response analysis exposes the potential for energy enhancement, albeit with inherent variations. Noteworthy is the revelation that manipulating the stiffness of the elastic foundation triggers shifts in the resonant frequency of the output voltage. Therefore, the proposed tunable elastic foundation system exhibits promising potential to engender versatile and adaptive phononic crystal configurations, thereby advancing the domain of PEH.

Band structure computation of two-dimensional and three-dimensional phononic crystals using a finite element-least square point interpolation method

• FE-LSPIM proposed to calculate band structures of elastic waves in 2D and 3D models. • Shape functions in FE-LSPIM exploit advantages of mesh-free/finite element methods. • FE-LSPIM can achieve excellent accuracy, especially in high-frequency modes. • FE-LSPIM achieves higher accuracy when calculating band gaps for phononic crystals. In the present study, a finite element-least square point interpolation method (FE-LSPIM) is proposed for calculating the band structures of in-plane elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs). This method utilizes new shape functions by combining mesh-free shape functions and finite element shape functions to exploit the specific advantages of the mesh-free method and finite element method (FEM). As a result, FE-LSPIM inherits the completeness properties of the mesh-free method and the compatibility properties of FEM, and thus the solutions obtained tend to be more accurate. Indeed, according to our previous research, the present method obtains excellent accuracy, especially in the high-frequency domain. The proposed FE-LSPIM was combined with Bloch's theory and applied to compute the band gaps (BGs) for 2D PCs and 3D PCs in the present study, where several PCs were investigated to verify the high accuracy when computing the BGs. Numerical analysis showed that the proposed method can predict the BGs more precisely compared with the FEM and modified FEM.

In-plane elastic wave propagation in nanoscale periodic piezoelectric/piezomagnetic laminates

Owing to the piezoelectric/piezomagnetic effects, the band structures of elastic waves in nanoscale smart periodic structures can be tuned actively, which can provide some new thoughts for the designs and applications of the nanoscale wave devices. In this paper, taking the size-effect into account and based on the nonlocal theory, the equations of wave motion including the magneto-electro-mechanical coupling effect are derived and solved. In particular, the propagation of elastic waves in nanoscale periodic piezoelectric/piezomagnetic laminates is investigated. Both the normal and oblique incidences are considered. The localization factors and dispersion curves are computed by the transfer matrix method, and the transmission spectra of a finite structure are obtained by the stiffness matrix method. The results show that the wave propagation properities such as the band structures, the wave localization degree and the value of the cut-off frequency can be tuned by the piezoelectric/piezomagnetic effects and the nanoscale size-effect.

Stable elastic wave band-gaps of phononic crystals with hyperelastic transformation materials

The elastic wave band structure in a phononic crystal (PC) is usually affected by the deformations in its soft constituent phase. In this work, hyperelastic transformation materials are proposed in the design of PCs in order to achieve stable elastic band-gaps that do not vary with deformation. It is demonstrated that one-dimensional PCs with a semi-linear soft phase can keep all elastic wave modes unchanged with respect to external deformations. However, only S-wave modes can be precisely retained in the PCs made of a neo-Hookean soft material. The theoretical results and the robustness of the proposed PCs are validated by numerical simulations.

Acoustic band structures and time reversal of elastic waves in two- and three-dimensional phononic crystals

The present study aims to investigate experimentally the acoustic band structures of elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs) consisting of periodic square arrays of cylindrical air holes in a solid matrix made of cubic polyoxymethylene. In the 2D PC, the first band gap appeared at frequencies between 0.50 and 0.77 MHz, and the second pass band at frequencies between 0.77 and 1.17 MHz. On the other hand, the 3D PC exhibited a wider band gap extending from a frequency of 0.44 MHz with noise level transmission at a frequency around 0.93 MHz. The time compressions of the time-reversed waves through the 2D and the 3D PCs were also obtained by using a time reversal process. The 2D PC exhibited better time compression than the 3D PC in the time-reversed waves, consistent with the fact that the time compression achieved by time reversal is sensitive to the number of frequency information grains.

Band structures of elastic waves in two-dimensional eight-fold solid–solid quasi-periodic phononic crystals**Project supported by the National Natural Science Foundation of China (Grant Nos. 11272043 and 10902012) and the Project-sponsored by SRF for ROCS, SEM.

Combined with the supercell method, band structures of the anti-plane and in-plane modes of two-dimensional (2D) eight-fold solid–solid quasi-periodic phononic crystals (QPNCs) are calculated by using the finite element method. The influences of the supercell on the band structure and the wave localization phenomenon are discussed based on the modal distributions. The reason for the appearance of unphysical bands is analyzed. The influence of the incidence angle on the transmission spectrum is also discussed.

Elastic Wave Band Structures and Defect States in a Periodically Corrugated Piezoelectric Plate

The band structures of shear horizontal (SH) waves in a periodically corrugated piezoelectric plate (PCPP) are studied by using the supercell plane wave expansion (SC-PWE) method. The effect of plate symmetry on the defect state caused by a defect in the plate is investigated in detail. The PCPPs with different types of symmetry give rise to different kinds of band gaps and the associated defect states. The increase of defect size lowers the frequency of defect bands, and it can be used to tune the narrow-passband frequencies in acoustic band gaps. Symmetry breaking is also introduced by reducing the lower corrugation depth of the PCPP. Results show that symmetry breaking leads to both the appearance and disappearance of new kinds of gaps and the corresponding defect bands in these gaps.

FINITE ELEMENT ANALYSIS OF THE INTERFACE/SURFACE EFFECT ON THE ELASTIC WAVE BAND STRUCTURE OF TWO-DIMENSIONAL NANOSIZED PHONONIC CRYSTALS

In this paper, an interface/surface element is formulated based on the Gurtin–Murdoch interface/surface elasticity theory for accounting the interface/surface effect on the elastic wave propagation in two-dimensional nanosized phononic crystals. The interface/surface element is subsequently incorporated into the finite element procedure for calculating the elastic wave band structure of two-dimensional nanosized phononic crystals with consideration of the interface/surface effect. Elastic wave band structures of two-dimensional phononic crystals comprising a square array of circular or elliptical cylindrical nanoholes embedded in an aluminum matrix are analyzed. Numerical results evidence that the interface/surface effect on the elastic wave band structure can be remarkable when the characteristic size reduces to nanometers.

Influence of parameter mismatch on the convergence of the band structures by using the Fourier expansion method

In the present paper, the convergence of the Fourier expansion method in calculating the band structures of elastic waves propagating in periodic composites is discussed. First, the convergence of the partial sum of a periodic function with discontinuous points is investigated by introducing three parameters to describe the overall error of a Fourier series and the Gibbs oscillation region. Second, the convergence of a product of two discontinuous functions is discussed based on two approximation methods, i.e., Laurent theory and the inverse formula, respectively. Finally, as an application, elastic waves propagating in one-dimensional layered periodic composite are studied. The band structures of the layered composites are obtained and some major influencing factors are discussed. Attentions are concentrated on the influences of the mismatch of mass density, shear modulus and filling ratio on the convergence of dispersion curves.

Wave Propagation Characteristics of One Dimensional Rod Phononic Crystal

The elastic wave band structures of the one dimensional rod phononic crystal are studied by the lumped-mass method. For the infinite periodic structure, the accuracy of numerical results is influenced by the number of discrete mass. The initial and stop frequecy of the first bandgap need different number of discrete mass to achieve calculation accuracy when two materials composed phononic crystal at different volume ratios. For the finite structure, the different arrangements make different width of the attenuation area at periodic load. The width of the bangap exhibits largely when the external load acts on the matrial with lower denstiy and elastic modulus in front of the higher density and elastic mudulus material.

Effect of interface/surface stress on the elastic wave band structure of two-dimensional phononic crystals

In the present Letter, the multiple scattering theory (MST) for calculating the elastic wave band structure of two-dimensional phononic crystals (PCs) is extended to include the interface/surface stress effect at the nanoscale. The interface/surface elasticity theory is employed to describe the nonclassical boundary conditions at the interface/surface and the elastic Mie scattering matrix embodying the interface/surface stress effect is derived. Using this extended MST, the authors investigate the interface/surface stress effect on the elastic wave band structure of two-dimensional PCs, which is demonstrated to be significant when the characteristic size reduces to nanometers.

Lumped-mass approach for the calculation of band structure of two-dimensional phononic crystals with high contrast of elastic constant

With each unit cell replaced by a system of finite freedoms of motion, two_dimensional phononic crystals can be simplified to an infinite discrete periodic system. Therefore, the elastic wave band structures of the two_dimensional phononic crystals can be calculated with a straightforward lumped_mass approach, whose computational cost is much lower than the well_known plane wave expansion(PWE) method. The numerical results of the two methods are in reasonable agreements. As the well_known Gibbs oscillations in the PWE can be eliminated with the lumped_mass method, this new approach is insensitive to the sharp variation of elastic constants on the interfaces inside the phononic crystals. Furthermore, the lumped_mass method can also be used to calculate the band structures of two_dimensional phononic crystals with arbitrary unit shapes easily.

Lumped-mass method on calculation of elastic band gaps of one-dimensional phononic crystals

With each unit cell replaced by finite cascaded massspring oscillators, 1D pho nonic crystals (PCs) can be simplified to an infinite periodic massspring chai n. Therefore, the elastic wave band structures of the 1D PCs can be calculated w ith a straightforward approach which we call lumpedmass method (LM). For compa rison, the band structures of the same 1D PC calculated with this method and the wellknown plane wave expansion method are both presented in this paper. The r esults of the two methods are in good agreement while the computation costs of L M method are much lower than that of PWE method.

Elastic wave scattering by periodic structures of spherical objects: Theory and experiment

We extend the multiple-scattering theory for elastic waves by taking into account the full vector character. The formalism for both the band structure calculation and the reflection and transmission calculations for finite slabs is presented. The latter is based on a double-layer scheme which obtains the reflection and transmission matrix elements for the multilayer slab from those of a single layer. As a demonstration of applications of the formalism, we calculate the band structures of elastic waves propagating in a three-dimensional periodic arrangement of spherical particles and voids, as well as the transmission coefficients through finite slabs. In contrast with the plane-wave method, the multiple-scattering approach exhibits advantages in handling specialized geometries (spherical geometry in the present case). We also present a comparison between theory and ultrasound experiment for a hexagonal-close-packed array of steel balls immersed in water. Excellent agreement is obtained.

Complex elastic wave band structures in three-dimensional periodic elastic media

Propagation of elastic waves can be completely inhibited, irrespective of propagation directions, if oscillation frequencies of elastic waves fall inside elastic wave band gaps—three-dimensional frequency stop bands of elastic waves generated in three-dimensional periodic elastic materials. The three-dimensional linear vector equation of motion is transformed into a matrix eigensystem and solved numerically by the plane wave expansion method. Existence of elastic wave band gaps is theoretically demonstrated for the face centered cubic lattice structure based on isotropie materials. The example structure is spherical tungsten scatterers embedded in a polyethylene matrix. Complex elastic wave band structures were also computed for tunneling evanescent waves at symmetric points at X and L points.

Suppression and enhancement of elastodynamic radiation from a point source load in elastic wave band structures

We show that propagation of elastic waves can be completely inhibited, irrespective of propagation directions, if oscillation frequencies of elastic waves fall inside elastic wave band gaps—three-dimensional frequency stop bands of elastic waves generated in three-dimensional periodic elastic materials. We also develop a classical theory for the calculation of radiated fields and energy from a point source load located in three-dimensional periodic elastic media by incorporating the plane wave method, dyadic Green’s function, Poynting theorem, and tetrahedron k-space integration. The fcc structure of tungsten spheres in polyethylene background is studied as an example. The resulting frequency spectra indicate the total inhibition of energy transmission in the elastic wave band gaps and unusual enhancement near the band edges. In addition, the computed data suggest the strong dependency of radiation spectra on the source position in the periodic elastic materials.