In-plane Elastic Waves Research Articles

Identical band gaps in structurally re-entrant honeycombs.

Structurally re-entrant honeycomb is a sort of artificial lattice material, characterized by star-like unit cells with re-entrant topology, as well as a high connectivity that the number of folded sheets jointing at each vertex is at least six. In-plane elastic wave propagation in this highly connected honeycomb is investigated through the application of the finite element method in conjunction with the Bloch's theorem. Attention is devoted to exploring the band characteristics of two lattice configurations with different star-like unit cells, defined as structurally square re-entrant honeycomb (SSRH) and structurally hexagonal re-entrant honeycomb (SHRH), respectively. Identical band gaps involving their locations and widths, interestingly, are present in the two considered configurations, attributed to the resonance of the sketch folded sheets, the basic component elements for SSRH and SHRH. In addition, the concept of heuristic models is implemented to elucidate the underlying physics of the identical gaps. The phenomenon of the identical bandgaps is not only beneficial for people to further explore the band characteristics of lattice materials, but also provides the structurally re-entrant honeycombs as potential host structures for the design of lattice-based metamaterials of interest for elastic wave control.

Analytical extension of Finite Element solution for computing the nonlinear far field of ultrasonic waves scattered by a closed crack

Nonlinear scattering of ultrasonic waves by closed cracks subject to contact acoustic nonlinearity (CAN) is determined using a 2D Finite Element (FE) coupled with an analytical approach. The FE model, which includes unilateral contact with Coulomb friction to account for contact between crack faces, provides the near-field solution for the interaction between in-plane elastic waves and a crack of different orientations. The numerical solution is then analytically extended in the far-field based on a frequency domain near-to-far field transformation technique, yielding directivity patterns for all linear and nonlinear components of the scattered waves. The proposed method is demonstrated by application to two nonlinear acoustic problems in the case of tone-burst excitations: first, the scattering of higher harmonics resulting from the interaction with a closed crack of various orientations, and second, the scattering of the longitudinal wave resulting from the nonlinear interaction between two shear waves and a closed crack. The analysis of the directivity patterns enables us to identify the characteristics of the nonlinear scattering from a closed crack, which provides essential understanding in order to optimize and apply nonlinear acoustic NDT methods.

Band structure computation of in-plane elastic waves in 2D phononic crystals by a meshfree local RBF collocation method

In this paper, the band structures of in-plane elastic waves in two-dimensional (2D) phononic crystals are calculated by using a meshfree local radial basis functions (RBF) collocation method. In order to improve the stability of the local RBF collocation method, special techniques suggested in our previous work for anti-plane waves are further improved and extended for calculating the primary field quantities and their normal derivatives required by the treatment of the boundary conditions in the local RBF collocation method for computing the band structures of the in-plane elastic waves in 2D phononic crystals. The developed meshfree local RBF collocation method for the band structure calculations of in-plane elastic waves propagating in 2D phononic crystals is validated by using the corresponding numerical results obtained with the finite element method (FEM). The band structures of different material combinations or acoustic impedance ratios, different filling fractions, various lattice forms and scatterer shapes are computed numerically to show the accuracy and the efficiency of the meshfree local RBF collocation method for computing the band structures of in-plane elastic waves in 2D phononic crystals.

Wave motion analysis and modeling of membrane structures using the wavelet finite element method

In this study, we present an application of the B-spline wavelet on the interval element method to analyze the in-plane elastic wave motion in a two-dimensional membrane structure. In contrast to traditional polynomial interpolation in classical finite element methods, the scaling function at a certain scale is used to form the shape functions and to construct wavelet-based elements. Other numerical wavelet methods add the wavelets directly, whereas the element displacement field represented by the coefficients of wavelets expansions in the proposed method is transformed from wavelet space to physical space via the corresponding transformation matrix. Numerical experiments are presented to demonstrate the effects of in-plane wave propagation in intact/notched membranes, particularly the propagations of the primary wave, secondary wave, and Rayleigh wave. In order to ensure a comprehensive analysis of the problem, the responses of the membrane are simulated under broad-band and narrow-band excitation.

In-plane time-harmonic elastic wave motion and resonance phenomena in a layered phononic crystal with periodic cracks.

This paper presents an elastodynamic analysis of two-dimensional time-harmonic elastic wave propagation in periodically multilayered elastic composites, which are also frequently referred to as one-dimensional phononic crystals, with a periodic array of strip-like interior or interface cracks. The transfer matrix method and the boundary integral equation method in conjunction with the Bloch-Floquet theorem are applied to compute the elastic wave fields in the layered periodic composites. The effects of the crack size, spacing, and location, as well as the incidence angle and the type of incident elastic waves on the wave propagation characteristics in the composite structure are investigated in details. In particular, the band-gaps, the localization and the resonances of elastic waves are revealed by numerical examples. In order to understand better the wave propagation phenomena in layered phononic crystals with distributed cracks, the energy flow vector of Umov and the corresponding energy streamlines are visualized and analyzed. The numerical results demonstrate that large energy vortices obstruct elastic wave propagation in layered phononic crystals at resonance frequencies. They occur before the cracks reflecting most of the energy transmitted by the incoming wave and disappear when the problem parameters are shifted from the resonant ones.

Seismic response of buried metro tunnels by a hybrid FDM-BEM approach

In this work, we present a 2D elastodynamic model for the seismic response of subway tunnels embedded in a laterally inhomogeneous, multilayered geological region overlying the half-plane. To this end, a finite difference-boundary element methodology (FDM-BEM) is developed, with the latter method embedded in the former so as to capture near-site field effects. More specifically, the FDM is used for simulating in-plane elastic wave propagation from the underlying bedrock through the overlying soil deposits to the surface. A ‘box’ area is then defined within the original FDM mesh and contains lined tunnels. The ‘box’ is modeled by the BEM and its upper boundary coincides with the free surface of the geological deposit. This way, seismically-induced motions are imparted from the FDM mesh to the ‘box’ perimeter, so that the BEM may now be used to efficiently model the near-site layers which contain the tunnels. Verification studies are then successfully conducted for upward moving Gabor pulses, using the FDM alone, the present hybrid FDM-BEM and a hybrid FDM-finite element method formulation. Given that the FDM is defined in the time domain and the BEM in the frequency domain, the fast Fourier transform is used for linking these two constituent parts of the hybrid approach. This methodology is finally applied to a north–south geological cross-section of Thessaloniki, Greece, which contains two Metro tunnels placed directly below an important Roman-era monument known as the Arch of Galerius. Results are then given in the form of free-surface motions stemming from the Thessaloniki 5 July 1978 aftershock recorded at bedrock so as to establish the influence of the ongoing Metro line construction, now temporarily halted because of the economic crisis, on the free surface motions in the city centre where a number historical monuments besides the Arch of Galerius still survive.

Topological optimization of two-dimensional phononic crystals based on the finite element method and genetic algorithm

By using the finite element method and a "coarse to fine" two-stage genetic algorithm as the forward calculation method and the inverse search scheme, respectively, we perform both the unconstrained and constrained optimal design of the unit cell topology of the two-dimensional square-latticed solid phononic crystals (PnCs), to maximize the relative widths of the gaps between the adjacent energy bands of the PnCs. In the constrained optimizations, the maximization is subjected to the constraint of a predefined average density. In the numerical results, the variation patterns of the optimized structures with the order of the bandgap for both the out-plane shear and the in-plane mixed elastic wave modes are presented, and the effects of both the material contrast and the predefined average density on the obtained optimal structures are discussed.

In-plane elastic wave propagation and band-gaps in layered functionally graded phononic crystals

In-plane wave propagation in layered phononic crystals composed of functionally graded interlayers arisen from the solid diffusion of homogeneous isotropic materials of the crystal is considered. Wave transmission and band-gaps due to the material gradation and incident wave-field are investigated. A classification of band-gaps in layered phononic crystals is proposed. The classification relies on the analysis of the eigenvalues of the transfer matrix for a unit-cell and the asymptotics derived for the transmission coefficient. Two kinds of band-gaps, where the transmission coefficient decays exponentially with the number of unit-cells are specified. The so-called low transmission pass-bands are introduced in order to identify frequency ranges, in which the transmission is sufficiently low for engineering applications, but it does not tend to zero exponentially as the number of unit-cells tends to infinity. A polyvalent analysis of the geometrical and physical parameters on band-gaps is presented.

Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence

A Finite Element (FE) model is proposed to study the interaction between in-plane elastic waves and a crack of different orientations. The crack is modeled by an interface of unilateral contact with Coulombs friction. These contact laws are modified to take into account a pre-stress σ0 that closes the crack. Using the FE model, it is possible to obtain the contact stresses during wave propagation. These contact stresses provide a better understanding of the coupling between the normal and tangential behavior under oblique incidence, and explain the generation of higher harmonics. This new approach is used to analyze the evolution of the higher harmonics obtained as a function of the angle of incidence, and also as a function of the excitation level. The pre-stress condition is a governing parameter that directly changes the nonlinear phenomenon at work at the interface and therefore the harmonic generation. The diffracted fields obtained by the nonlinear and linear models are also compared.

The Propagation of In-Plane Waves in Multilayered Elastic/Piezoelectric Composite Structures with Imperfect Interface

The dispersive characteristic of in-plane elastic waves propagating through laminated piezoelectric phononic crystal with imperfect interface is studied in this paper. First, the transfer matrix method (TMM) and the Bloch theorem are used to derive the dispersion equation. Next, the dispersion equation is solved numerically and the dispersive curves are shown in Brillouin zone. The pass band and the stop band of in-plane wave propagating normal to the laminated periodic structure with spring imperfect interface are investigated. The effects of the spring or mass parameter are discussed.

Elastic Wave Localization in Layered Phononic Crystals With Fractal Superlattices

In this paper, localization phenomena of in-plane time-harmonic elastic waves propagating in layered phononic crystals (PNCs) with different fractal superlattices are studied. For this purpose, oblique wave propagation in layered structures is considered. To describe wave localization phenomena, the localization factor is applied and computed by the transfer matrix method. Three typical fractal superlattices are considered, namely, the Cantorlike fractal superlattice (CLFSL), the golden-section fractal superlattice (GSFSL), and the Fibonacci fractal superlattice (FFSL). Numerical results for the localization factors of CLFSL, GSFSL, and FFSL are presented and analyzed. The results show that the localization factor of a CLFSL exhibits an approximate similarity and band-splitting properties. The number of decomposed bandgaps of the GSFSL and FFSL follows the composition of the special fractal structures. In addition, with increasing fractal series, the value of the localization factor is enlarged. These results are of great importance for structure design of fractal PNCs.

Active elastodynamic cloaking

An active elastodynamic cloak destructively interferes with an incident time harmonic in-plane (coupled compressional/shear) elastic wave to produce zero total elastic field over a finite spatial region. A method is described which explicitly predicts the source amplitudes of the active field. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. Importantly, the active field generated by the sources vanishes in the far-field. In practice the infinite summations are clearly required to be truncated and the accuracy of cloaking is studied when the truncation parameter is modified.

Band gap structures in two-dimensional super porous phononic crystals

As one kind of new linear cellular alloys (LCAs), Kagome honeycombs, which are constituted by triangular and hexagonal cells, attract great attention due to the excellent performance compared to the ordinary ones. Instead of mechanical investigation, the in-plane elastic wave dispersion in Kagome structures are analyzed in this paper aiming to the multi-functional application of the materials. Firstly, the band structures in the common two-dimensional (2D) porous phononic structures (triangular or hexagonal honeycombs) are discussed. Then, based on these results, the wave dispersion in Kagome honeycombs is given. Through the component cell porosity controlling, the effects of component cells on the whole responses of the structures are investigated. The intrinsic relation between the component cell porosity and the critical porosity of Kagome honeycombs is established. These results will provide an important guidance in the band structure design of super porous phononic crystals.

Beaming of inplane elastic waves through a subwavelength channel with periodic corrugations

We present an evidence of the beaming phenomenon regarding inplane elastic waves in the context of the enhanced acoustical transmission. Our simulations found emission of a long-range beam through a single subwavelength channel surrounded by periodic structures. The space-time spectral analysis of the inplane displacement field revealed that Rayleigh surface waves principally mediate the energy transmission that leads to the formation of beams. Consequently, we suggest a resonance condition for the beaming in terms of the ratio of the wavelength to the period of the structures. Our findings will be expected to foster the application to ultrasonic devices.

Band structures and localization properties of aperiodic layered phononic crystals

The band structures and localization properties of in-plane elastic waves with coupling of longitudinal and transverse modes oblique propagating in aperiodic phononic crystals based on Thue–Morse and Rudin–Shapiro sequences are studied. Using transfer matrix method, the concept of the localization factor is introduced and the correctness is testified through the Rytov dispersion relation. For comparison, the perfect periodic structure and the quasi-periodic Fibonacci system are also considered. In addition, the influences of the random disorder, local resonance, translational and/or mirror symmetries on the band structures of the aperiodic phononic crystals are analyzed in this paper.

Wave localization in two-dimensional porous phononic crystals with one-dimensional aperiodicity

The localization properties of in-plane elastic waves propagating in two-dimensional porous phononic crystals with one-dimensional aperiodicity are initially analyzed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method in this paper. The band structures characterized by using localization factors are calculated for different phononic crystals by altering matrix material properties and geometric structure parameters. Numerical results show that the effect of matrix material properties on wave localization can be ignored, while the effect of geometric structure parameters is obvious. For comparison, the periodic porous system and Fibonacci system with rigid inclusion are also analyzed. It is found that the band gaps are easily formed in aperiodic porous system, but hard for periodic porous system. Moreover, compared with aperiodic system with rigid inclusion, the wider low-frequency band gaps appear in the aperiodic porous system.

Ultrawide phononic band gap for combined in-plane and out-of-plane waves

We consider two-dimensional phononic crystals formed from silicon and voids, and present optimized unit-cell designs for (1) out-of-plane, (2) in-plane, and (3) combined out-of-plane and in-plane elastic wave propagation. To feasibly search through an excessively large design space (~10(40) possible realizations) we develop a specialized genetic algorithm and utilize it in conjunction with the reduced Bloch mode expansion method for fast band-structure calculations. Focusing on high-symmetry plain-strain square lattices, we report unit-cell designs exhibiting record values of normalized band-gap size for all three categories. For the case of combined polarizations, we reveal a design with a normalized band-gap size exceeding 60%.

Elastic Wave Localization in Two-Dimensional Phononic Crystals with One-Dimensional Aperiodicity

The band structures of in-plane elastic waves propagating in two-dimensional phononic crystals with one-dimensional aperiodicity are analyzed in this paper. The localization of wave propagation is discussed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method. By treating the aperiodicity as the deviation from the periodicity in a special way, two kinds of aperiodic phononic crystals that have Thue-Morse and Rudin-Shapiro sequence in one direction and translational symmetry in the other direction are considered. The transmission coefficients based on eigenmode match theory are also calculated and the results show the same behaviors as the localization factor does. In the case of Thue-Morse and Rudin-Shapiro structures, the band structures of Thue-Morse sequence exhibit similarities with quasi-periodic sequence not present in the results of Rudin-Shapiro sequence.

Modulation of the bandgaps of in-plane elastic waves by out-of-plane wavenumber in thepiezoelectric composite structures

We study theoretically the modulation of bandgaps of in-plane elastic waves by theout-of-plane wavenumber that is parallel to the piezoelectric material rod axis inpiezoelectric composite structures. The dependence of bandgaps of in-plane elasticwaves on the out-of-plane wavenumber and the filling fraction are calculatedwith and without the composite’s piezoelectricity. The in-plane bandgaps can beopened (or closed) and the in-plane wave modes can be tuned by controlling thepropagation of out-of-plane elastic waves. The in-plane bandgap in the low-frequencyrange can be easily obtained by introducing an out-of-plane wavenumber, and isalmost independent of the composite’s piezoelectricity. Meanwhile, the widths andstarting frequencies of in-plane bandgaps in a relatively high-frequency range canbe modulated by the filling fraction and the composite’s piezoelectricity. Thepiezoelectric composite structure is very useful in the design of ultrasonic transducers toavoid the invalid modes of in-plane elastic waves without affecting the workingmodes corresponding to the out-of-plane wavenumber. The method to calculatethe total transmission spectrum of elastic waves in this paper is useful to studypiezoelectric composite structures with finite length in engineering applications.

Elastic wave localization in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity

The band structures of in-plane elastic waves propagating in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity are analyzed in this paper. The localization of wave propagation is discussed by introducing the concept of the localization factor, which is calculated by the plane-wave-based transfer-matrix method. By treating the random disorder and aperiodicity as the deviation from the periodicity in a special way, three kinds of aperiodic phononic crystals that have normally distributed random disorder, Thue–Morse and Rudin–Shapiro sequence in one direction and translational symmetry in the other direction are considered and the band structures are characterized using localization factors. Besides, as a special case, we analyze the band gap properties of a periodic planar layered composite containing a periodic array of square inclusions. The transmission coefficients based on eigen-mode matching theory are also calculated and the results show the same behaviors as the localization factor does. In the case of random disorders, the localization degree of the normally distributed random disorder is larger than that of the uniformly distributed random disorder although the eigenstates are both localized no matter what types of random disorders, whereas, for the case of Thue–Morse and Rudin–Shapiro structures, the band structures of Thue–Morse sequence exhibit similarities with the quasi-periodic (Fibonacci) sequence not present in the results of the Rudin–Shapiro sequence.