Localization Of Elastic Waves Research Articles

Controllable manipulation of surface waves via a ferromagnetic elastic metasurface

In this paper, we study the propagation of surface waves on an A-shape metasurface, whose structure of the unit-cell is composed regularly of an A-shape surface column (made of steel), a thin magnetorheological plate and a soft thick rubber base. On the basis of the commercial program COMSOL Multiphysics, finite element method (FEM) analyses are carried out. The localization of elastic waves surrounding the surface allows surface waves to be distinguished among various complex wave modes. By adjusting the geometry, the A-shape surface column can be transformed from snowflake to A-shape. Owing to the C3v symmetry system, a Dirac cone of surface wave is observed in a snowflake metasurface. When the surface column changes to A-shape, the C3v symmetry is broken, making the Dirac cone vanish and a new band gap of surface waves arise. The band gap can be opened or closed by changing a geometric parameter, which determines the shape of the surface column changing from gadarene A-shape to snowflake or even upturned A-shape. Furthermore, we discover that although the topological phases of the gadarene A-shape and upturned A-shape metasurfaces are opposite, they share overlapped band gaps. Based on this phenomenon, we design a topological insulator for surface waves, whose super-cell is composed of gadarene A-shape units and upturned A-shape units. Through the calculation of the super-cell band structure, we observe the topologically protected interface states (TPISs). Subsequently, we design several waveguides whose paths are controllable. Up to now, most of the existing metasurfaces are made of passive materials, whose properties cannot be adjusted in real time, making them inconvenient for engineering applications when external conditions are changed. Therefore, we use tunable magnetorheological elastomer (MRE) to control the properties of the topological insulators. The stiffness of the surface layer can be converted by applying a magnetic field, which makes the topological metasurface controllable. These intelligent topological insulators may find important applications in the design of surface acoustic wave (SAW) devices.

Geometric localization of waves on thin elastic structures.

We consider the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a singly curved shell as concrete examples. Previous studies on related problems have broadly focused on the localization of flexural waves on such structures. Here, using the semiclassical WKB approximation for multicomponent waves, we show that in addition to flexural waves, extensional and shear waves also form localized, bound states around points where the absolute curvature of the structure has a minimum. We also see excellent agreement between our numerical experiments and the semiclassical results, which hinges on the vanishing of two extra phases that arise in the semiclassical quantization rule. Our findings open up novel ways to fine-tune the acoustic and vibrational properties of thin elastic structures and raise the possibility of introducing new phenomena not easily captured by effective models of flexural waves alone.

Size and Temperature Effects on Band Gap Analysis of a Defective Phononic Crystal Beam

In this paper, a new defective phononic crystal (PC) microbeam model in a thermal environment is developed with the application of modified couple stress theory (MCST). By using Hamilton’s principle, the wave equation and complete boundary conditions of a heated Bernoulli–Euler microbeam are obtained. The band structures of the perfect and defective heated PC microbeams are solved by employing the transfer matrix method and supercell technology. The accuracy of the new model is validated using the finite element model, and the parametric analysis is conducted to examine the influences of size and temperature effects, as well as defect segment length, on the band structures of current microbeams. The results indicate that the size effect induces microstructure hardening, while the increase in temperature has a softening impact, decreasing the band gap frequencies. The inclusion of defect cells leads to the localization of elastic waves. These findings have significant implications for the design of microdevices, including applications in micro-energy harvesters, energy absorbers, and micro-electro-mechanical systems (MEMS).

Localization of elastic waves in one-dimensional detuned phononic crystals with flexoelectric effect

ABSTRACT Although forbidden band effect in perfectly periodic phononic crystals (PC) is very attractive, random disordered (i.e. detuning) phenomenon is inevitable in engineering processing, thus exploring the effect of detuning on the wave characteristics of PC becomes a necessity. In this study, fundamental governing equations and boundary conditions are derived from the principle of virtual work. Wave characteristics and localization factor of one-dimensional (1D) detuned nano-PC are investigated based on the transfer matrix method, with flexoelectric effect duly accounted for. Subsequently, with nano-PC taken for illustration, forbidden band properties and localization factor of 1D elastic waves in harmonic and detuned states are systematically characterized. It is demonstrated that localization factor can characterize the energy band structure of 1D PC perfectly. Flexoelectric effect tends to widen the width of forbidden band with increasing detuning, and detuning is linearly related to the bandwidth. The forbidden band is more sensitive to flexoelectric coefficient detuning than thickness detuning. The research results provide useful theoretical guidance for designing high-frequency nanoscaled devices with the function of filtering based on band gap effect of nano-PC.

Extremely low frequency wave localization via elastic foundation induced metamaterial with a spiral cavity

We proposed a metamaterial which exhibits elastic wave localization at extremely low frequencies. First, we opened an extremely low bandgap via elastic foundations. Subsequently, we investigated wave localization by imposing normal defect, which is widely used to capture waves in conventional wave localization systems. However, there were limitations: wave localization was not achieved when a weak bandgap is generated, and the operating frequency of localization is still in the upper part of the bandgap. To overcome wave localization via the normal defect, we proposed a novel metamaterial with a spiral cavity which can tune the resonating frequency depending on the length of the spiral path. By imposing on the spiral cavity inside the elastic foundation-induced metamaterial, we can shift the resonating frequency of the cavity down. Finally, we carried out wave simulations, not only to support the previous eigenfrequency study for the supercell, but also to verify that the finite-size metamaterial can also achieve wave localization at the extremely low frequencies. Through wave simulations, we could observe wave localization even at 77.3 Hz, which is definitely the lower part of the extremely low bandgap.

Space–time wave localization in electromechanical metamaterial beams with programmable defects

A relevant feature for various applications of phononic crystals and metamaterials is the wave localization and guiding using defects introduced into a periodic array. In this context, we present a flexible localization of elastic waves in piezoelectric metamaterial beams with space–time-modulated defects. The vibration energy, originally confined in a point defect, is gradually manipulated in space and time as the inductances of the shunt circuits of subsequent unit cells are properly controlled, perfectly inducing energy localization at the final defect position. This wave localization by programmable defects is also applied to a two-dimensional triangular lattice composed by piezoelectric metamaterial beams, which illustrates the potential and versatility of the proposed strategy in more complex systems. The space–time modulation of point defects presented in this work opens new possibilities for programmable wave localization, and hence, manipulation of vibration energy or information using smart metamaterials, with applications that may involve guiding, sensing, monitoring and energy harvesting.

Double defects-induced elastic wave coupling and energy localization in a phononic crystal

This study aims to investigate elastic wave localization that leverages defect band splitting in a phononic crystal with double defects through in-depth analysis of comparison of numerical and experimental results. When more than one defect is created inside a phononic crystal, these defects can interact with each other, resulting in a distinctive physical phenomenon from a single defect case: defect band splitting. For a phononic crystal consisting of circular-hole type unit cells in a thin aluminum plate, under A0 (the lowest antisymmetric) Lamb waves, both numerical simulations and experiments successfully confirm the defect band splitting phenomenon via frequency response functions for the out-of-plane displacement calculated/measured at the double defects within a finite distance. Furthermore, experimental visualization of in-phase and out-of-phase defect mode shapes at each frequency of the split defect bands is achieved and found to be in excellent agreement with the simulated results. Different inter-distance combinations of the double defects reveal that the degree of the defect band splitting decreases with the increasing distance due to weaker coupling between the defects. This work may shed light on engineering applications of a multiple-defect-introduced phononic crystal, including broadband energy harvesting, frequency detectors, and elastic wireless power transfer.

Transport and Localization of Elastic Waves in Two‐Dimensional Fractured Media: Consequences on Scattering Attenuation

AbstractWe present numerical simulations of elastic wave transport in two‐dimensional fractured media. Natural fracture systems, following a power‐law length scaling, are modeled by the discrete fracture network approach for geometrically representing fracture distributions and the displacement discontinuity method for mechanically computing fracture‐wave interactions. The model is validated against analytical solutions for wave reflection, transmission, and scattering by single fractures, and then applied to solve the wavefield evolution in synthetic fracture networks. We find that the dimensionless angular frequency ῶ = ωZ/κ plays a crucial role in governing wave transport, where ω, Z, and κ are the angular frequency, seismic impedance, and fracture stiffness, respectively. When ῶ is smaller than the critical frequency ῶc (≈5), waves are in the extended mode, either propagating (for small ῶ) or diffusing by multiple scattering (for intermediate ῶ); as ῶ exceeds ῶc, waves become trapped, entering either the Anderson localization regime (kl* ≈ 1) in well‐connected fracture systems or the weak localization regime (kl* > 1) in poorly‐connected fracture systems, where k is the incident wavenumber and l* is the mean free path length. Consequently, the inverse quality factor Q−1 scales with ῶ obeying a two‐branch power‐law dependence, showing significant frequency dependence when ῶ < ῶc and almost frequency independence when ῶ > ῶc. Furthermore, when ῶ < ῶc, the wavefield exhibits a weak dependence on fracture network geometry, whereas when ῶ > ῶc, the fracture network connectivity has an important impact on the wavefield such that strong attenuation occurs in well‐connected fracture systems.

Elastic wave localization and energy harvesting defined by piezoelectric patches on phononic crystal waveguide

The application of piezoelectric effect to localize elastic wave and collect energy has been a hot research field in recent years. In this paper, we designed a phononic crystal (PnC)-based system with a piezoelectric waveguide to realize wave localization. (PnC) is one representative example of a metamaterial, it is constituted by a periodic repetition of unit cells. The most important feature of PnC is to forbid wave propagate in some special frequency, people called it bandgap. In our research, we introduce line defect waveguide and piezoelectric waveguide in the bandgap. By adding piezoelectric patches in the waveguide of PnC structure, the frequency of waveguide mode can be raised from 2240 Hz to 2265 Hz. The frequency difference between the two waveguides helps us create a wave localization in the position of piezoelectric patches. We also find that the localization performs best when the frequency is 2257 Hz. Besides, the electro-elastic coupling is studied in this paper, the coupling would become stronger when the two piezoelectric patches get closer, then we can get a vibration peak in strong coupling, it can help us localize wave if we need a very strong vibration in a point or a small position. On the contrary, the coupling would become weaker when the two piezoelectric patches get farther, and then we can obtain a wide localization position with the same amplitude of vibration between two piezoelectric patches, it can help us localize wave if we need the same vibration in a relatively wide area. We also investigate the relationship between power and the resistance in the circuit, results show that the maximum power appears in 103Ω of all the three cases. The research of wave localization and energy harvesting can give us a new idea to design many vibration and energy-localization devices.

A graded phononic crystal with decoupled double defects for broadband energy localization

Several research efforts have been devoted to exploring defect modes of uniform phononic crystals (PnCs) with single or double defects. However, they have difficulties in achieving broadband energy localization under realistic environments. This study thus proposes a new design concept of a graded PnC with double defects for broadband energy localization. A graded PnC refers to a PnC built by varying one or more design variables of its lattices. The double defects are introduced to be sufficiently far from each other to make them decoupled; each of the defects behaves as an independent single defect. Unlike a uniform PnC, the graded PnC enables to separately design defect band frequencies corresponding to each of the double defects by changing a degree of the variation in the height of the circular stub. The graded PnC with double defects can broaden the bandwidth for elastic wave localization.

Elastic wave localization and harvesting using double defect modes of a phononic crystal

Phononic crystals (PnCs) have been utilized to amplify the amount of input energy transferred to a piezoelectric energy harvesting (PEH) device by manipulating elastic wave propagation. When introducing a defect that has material properties and/or geometry different from a unit cell, mechanical resonance of the defect leads to localizing elastic waves inside the defect. This is called a defect mode. Several prior studies have explored a defect mode for PEH purpose; however, they have focused only on a single defect. When introducing an additional defect into a PnC, the coupling between two defects leads to splitting the defect band. Incorporating such split defect band phenomena into PEH can potentially widen frequency bandwidth and realize broadband energy harvesting. Thus, this study newly proposes a PnC-based PEH system that uses double defect modes under elastic waves. In particular, this study examines how an electrical circuit connection (i.e., Independent, Series, or Parallel) between two PEH devices attached on each defect affects PEH performances. Key findings from this study include (1) the shift in split defect bands as well as harvesting performance varies with the type of electrical circuit connection; (2) split defect band phenomena amplify the output electric power at double resonance peaks in an Independent connection; and (3) despite displacement amplification at certain resonance frequencies, no peak output voltage occurs due to the current source direction (for Series connections) and voltage cancellation (for Parallel connections). These key findings can provide design and selection guidelines for electrical circuit configurations between double defects for enhanced PEH.

Active localization of wave propagation in elastic beams using periodic placement of piezoelectric actuator/sensor pairs

A novel strategy is proposed to investigate the active localization of elastic waves in beams with periodically placed piezoelectric actuator/sensor pairs under random feedback control gains. The randomly disordered active stiffness is obtained and periodically added to the beam. The localization factor is introduced and defined by using the eigenvalues of the transfer matrices. From this investigation, it is found that the active localization of elastic waves in beams can be realized and the propagation of elastic waves in certain designated frequency ranges can be actively controlled. Consequently, the feedback control gains can be intentionally designed to be random so that the frequency band-gaps can be actively broadened and the ability for the structural vibration isolation and noise reduction can also be enhanced.

Spatial localization of linear elastic waves in composite materials with defects

AbstractWe study the phenomenon of a spatial localization of elastic waves in periodic composite materials with local defects. The wave spectrum in heterogeneous composite solids includes pass and stop frequency bands. If the frequency of the signal falls within a stop band, the group velocity vanishes and the wave attenuates exponentially. In such a case, a local perturbation of the microstructure may lead to the localization of the wave energy in the vicinity of the defect. Longitudinal tension‐compression waves in a layered composite and transverse antiplane shear waves in a unidirectional fibrous composite are considered. Local perturbations of the density and of the volume fractions of the components are taken into account. The analysis is based on the transfer‐matrix method and on the plane‐wave expansions method. As the result, the frequencies of the wave localization and the corresponding attenuation factors are determined.

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.

Thin Film Exfoliation in the Zone of Trapped Elastic Wave Localization

AbstractIn modern constructions, thin‐layer coats are often used as protecting or strengthening elements. Deformations of such constructions may cause significant stresses on the interface between the base and the coat because of the difference in their physical mechanical properties, which leads to the destruction or detachment of the cover. The impact of static or shock loads on the emergence and development of delamination in multilayer constructions has been studied fairly well, but much less is known about destructions of this type under nonstationary (vibrational) loads. The interest in the latter is caused by the fact that even small variable actions may cause the localization of oscillations in a neighborhood of inhomogeneities (such as inclusions, defects, construction elements, etc.) [1] (and lead to the emergence and growth of defects, which strongly affects the reliability and functionality of the whole construction. The possibility of the localization of oscillations near initial delamination and the impact of localization on the growth of the delamination are studied and a model for analyzing conditions under which a detachment zone grows or ceases to grow is suggested. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dispersion and localization of elastic waves in materials with microstructure

This paper considers the interaction of elastic waves with materials with microstructure. The paper presents a mathematical model of elastic waves within a lattice system incorporating rotational motions and interaction between different lattice elements through elastic links. The waves are dispersive and the lattice system itself is heterogeneous, i.e. the elastic stiffness and/or mass are non-uniformly distributed. For such systems, one can identify stop bands, representing the intervals of frequencies of waves, which become evanescent and cannot propagate through the structure. Filtering properties of such lattices are studied in this paper. Defect modes are created by removing a periodic array of elastic links, which leads to localization within a macro-cell. Special attention is given to the evaluation of the effective group velocities and to the study of standing waves within the system. Analytical estimates are accompanied by numerical simulations and analysis of dispersion surfaces. We also consider an example showing the focusing and the creation of an image point by a flat elastic ‘lens’ formed from a finite micropolar lattice system.

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.

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.

Wave propagation and localization in randomly disordered layered composites with local resonances

Localization of elastic waves, both in-plane and anti-plane waves, propagating obliquely or perpendicular to periodic and randomly disordered layered composites with and without local resonances is investigated. The transmission coefficient and the localization factor are determined by means of the transfer matrix method. Compared with the corresponding binary structures without local resonances, new phenomenon is found in the periodically layered fourfold composite structure with local resonances.

Wave localization in randomly disordered layered three-component phononic crystals with thermal effects

In this paper, the propagation and localization of elastic waves in randomly disordered layered three-component phononic crystals with thermal effects are studied. The transfer matrix is obtained by applying the continuity conditions between three consecutive sub-cells. Based on the transfer matrix method and Bloch theory, the expressions of the localization factor and dispersion relation are presented. The relation between the localization factors and dispersion curves is discussed. Numerical simulations are performed to investigate the influences of the incident angle on band structures of ordered phononic crystals. For the randomly disordered ones, disorders of structural thickness ratios and Lame constants are considered. The incident angles, disorder degrees, thickness ratios, Lame constants and temperature changes have prominent effects on wave localization phenomena in three-component systems. Furthermore, it can be observed that stopbands locate in very low-frequency regions. The localization factor is an effective way to investigate randomly disordered phononic crystals in which the band structure cannot be described.