Complex Dispersion Curves Research Articles

3D complex dispersion curves and attenuation characteristics based on Drude-Lorentz oscillators for Lamb wave in various metal-piezoelectric composites

Given that many micro-nano piezoelectric acoustic devices operate at very high frequencies, the dissipation caused by metal electrodes significantly affects their performance (e.g., quality factor), but these dissipation characteristics cannot be explained by conductivity at high frequencies. This study uses the Drude-Lorentz oscillator model, incorporating the frequency-dependent dielectric properties of metals, which in physics refer to electron oscillations at high frequencies, to investigate the three-dimensional (3D) complex dispersion curves and attenuation characteristics of Lamb waves in metal-piezoelectric composites. Five commonly used electrode metals (Pt, Al, Ag, Au, Cu) are analyzed to reveal the widespread attenuation characteristics. The Multidimensional Moduli Ratio Convergence Method (MMRCM) is employed, which utilizes the convergence and divergence of the moduli ratio to accurately locate zeros of complex dispersion equations. Meanwhile, multidimensional scanning is adopted to ensure comprehensive identification of minima moduli points. Two primary attenuation characteristics are identified: (1) attenuation trends related to the real part of the wavenumber for different branches, and (2) significant jumps in attenuation due to mode shape conversions in metals with veering regions. Furthermore, a size-dependent attenuation characteristic is observed, showing a quadratic increase in attenuation as the composite structure’s total thickness decreases. These findings provide crucial insights for optimizing the design and performance of micro-nano devices where precise control over wave attenuation and dispersion is essential.

On the bandgap mechanism of periodic acoustic black holes

Acoustic black holes (ABHs) are primarily intended to eliminate the propagation of bending waves in structures although the same physical principles have been used to reduce the propagation of acoustic waves in ducts. The latter are usually referred to as sonic black holes (SBHs). ABHs (and also SBHs) only work well above a certain cut-on frequency which depends on their size compared to the incident wavelength. This limits their effectiveness to the high frequency range. To overcome this problem, several works have proposed the design of periodic ABHs to generate stopbands for low frequencies and improve their operating range. The purpose of this communication is to shed some light on the nature of such bandgaps. The k(ω) method is used to calculate the complex dispersion curves of various periodic ABH configurations and it is shown that, contrary to what is suggested in many works, the generation of bandgaps is essentially due to Bragg scattering and not to local resonances for embedded ABHs, although the latter play a role in some cases. The opposite occurs for additive ABHs. It is also observed that the complex dispersion curves of periodic ABHs present an intriguing behaviour compared to those usually found in metamaterials, since they tend to disappear at high frequencies, where the local wave intensity is stronger. An explanation of all these facts is given in the case of a periodic SBH, single-leaf and double-leaf embedded ABHs, and pillar and single-leaf additive ABHs.

Periodic additive acoustic black holes to absorb vibrations from plates

This paper first suggests a composite structure with periodic additive acoustic black holes (ABH) to absorb vibrations from plates. Different from conventional embedded ABHs, this additive configuration no longer compromises structural integrity. The suggested periodic structure forms a compact acoustic functional material. To analyze the vibration behavior of this coupled structure, we establish a theoretical model employing the Gaussian expansion method (GEM) and the nullspace method (NSM), and it is validated through finite element simulations. Subsequently, we develop a two-dimensional wave and Rayleigh–Ritz method (WRRM) to compute complex dispersion curves, with the imaginary part predicting absorption capability. Our findings reveal that the local resonance eigenmodes of the ABH plate with high loss factors dominate the damping effect, overshadowing the role of Bragg scattering. Furthermore, the introduction of a damping layer proves highly effective in absorbing local vibrations across a wide frequency band. Moreover, we then conduct parametric analyses on various aspects, including ABH order, central thickness, ABH plate thickness, and ABH radius. Notably, the latter two parameters exert a significant influence on damping performance, primarily due to the added mass they introduce. Finally, we examine a finite plate consisting of a 4 × 4 cell arrangement. This composite structure demonstrates substantial modal loss factors when local modes are activated. Moreover, in forced vibration tests, the additive ABHs prove highly effective in mitigating vibrations within the host plate.

Complex dispersion analysis of true and pseudo surface waves propagating in two-dimensional viscoelastic seismic metamaterials

In this paper, the wave characteristics of true surface waves (TSWs) and pseudo surface waves (PSWs) in two-dimensional (2D) seismic metamaterials are explored. The 2D seismic metamaterial is formed by placing periodic concrete pillars on the surface of the soil substrate. The complex dispersion curves of the seismic metamaterial are numerically calculated by the ω–kapproach. The Kelvin–Voigt model is adopted to describe the material viscoelasticity. The results show that not only the TSW modes but also the PSW modes should be taken into account to evaluate the mitigation capacity of the seismic metamaterials. Due to the introduction of material viscoelasticity, the TSWs and PSWs convert from the propagative mode to the evanescent mode. When the PSW modes propagate along the substrate surface, their energy gradually leaks into the bulk of the substrate, leading to their transmission coefficient smaller than that of the TSW modes. Compared to the concrete pillars on the substrate surface, the viscoelasticity of the soil substrate plays a more important role in surface wave reduction. Moreover, the wave propagating length in the seismic metamaterials can be predicted by using complex dispersion relations.

Vibration damping by periodic additive acoustic black holes

Embedded acoustic black holes (ABHs) have demonstrated significant efficiency in vibration mitigation. However, their practical application is limited due to their inherent low strength. To address this issue, this paper introduces an alternative approach of additive ABHs, which are periodically mounted on the host structure to simultaneously enhance structural rigidity and damping effects. The Gaussian expansion method (GEM) is employed to model the ABH beam and base beam within the framework of Timoshenko beam theory. The coupling between the two substructures is achieved using the nullspace method (NSM) based on the interface coupling condition. Additionally, the wave and Rayleigh–Ritz method (WRRM) is utilized to handle the Bloch-Floquet periodic boundary conditions. The complex dispersion curves are obtained using the k(ω) method, where the imaginary part represents the attenuating waves in space. The accuracy of the proposed model is validated against finite element simulations for the unit cell modes and dispersion curves. The results indicate that local resonance dominates the behavior, overshadowing the Bragg scattering effect. Furthermore, the inclusion of a damping layer results in a strong damping effect across a wide frequency band. Parametric studies reveal that larger lattice constant and increased uniform portion thickness of the ABH contribute to better damping performance, albeit with an associated increase in the added mass of the ABH component. Finally, the damping capability of finite periodic beams with 5 cells is investigated, assessing modal loss factor (MLF), transmissibility, and forced vibration shapes. The findings demonstrate that the proposed additive ABH approach opens up new possibilities for engineering applications.

Sound waves in continuum models of periodic sonic black holes

In this work we address the study of continuum models of periodic sonic black holes (SBHs) at duct terminations. A conventional SBH combines a decreasing power-law profile with a wall admittance in such a way that sound waves entering the SBH slow down, while their amplitude increases and their wavelength decreases as they approach the duct termination. In an ideal scenario, the wave is trapped and dissipated inside the SBH so that no reflection occurs. Instead of dealing with conventional SBHs, we are interested here in periodic SBHs. These have the potential to extend the application range of SBHs below the cut-on frequency thanks to the formation of stopbands, among other significant aspects. First, the case of an SBH cell placed between semi-infinite ducts is analyzed and it is shown how it reduces not only sound reflection but also sound transmission. The acoustic pressure inside the SBH is governed by a modified Webster equation that is solved in weak form using a basis of Gaussian functions. The dependence of the reflection, transmission and absorption coefficients for different SBH parameters and boundary conditions is analyzed in detail. Next, the case of an infinite periodic SBH lattice is considered. The nullspace method (NSM) is used to impose the periodic boundary conditions of the problem. Complex dispersion curves are computed and the separate roles played by the SBH power-law profile and wall admittance on the dispersion curves and pressure distribution inside the cells are examined. Bragg scattering turns out to be the main mechanism behind bandgap formation and wall admittance is essential to achieve the SBH effect. A parametric study follows, showing the influence of the damping, the residual radius, the SBH order and its length on the imaginary part of the complex dispersion curves. Finally, the reflection, transmission and absorption coefficients for a finite 3-cell SBH are investigated. Although most of the research on SBHs has focused on practical application designs, it is believed that a thorough understanding of the performance of continuum models of SBHs could be very useful for the former.

Non-Fourier thermal wave in 2D cellular metamaterials: From transient heat propagation to harmonic band gaps

Tailoring the nano/microarchitecture of advanced materials is a new root for manipulating their behavior to attain unparalleled multifunctional properties, including guided transitional waves. Elastic , acoustic, and electromagnetic wave responses in architected materials have extensively been studied, while only few studies have been allocated to the thermal wave propagation in these engineered materials. In this article, we explore the propagation of thermal waves in cellular metamaterials as a category of rationally-designed architected materials. First, the propagation of an instantaneous heat excitation through a two-dimensional (2D) cellular medium is investigated in steady-state and transient modes by Fourier and non-Fourier (Cattaneo-Vernotte and dual-phase-lag) heat conduction theories. The effects of relative density and microarchitecture of the cellular domain on the heat propagation characteristics are investigated, connecting these traits to the effective thermal conductivity of the cellular domain using an asymptotic homogenization method. Second, by implementing the Bloch theorem for non-Fourier heat conduction models, complex dispersion curves are obtained for five alternative cellular architectures with Archimedean tilings. The cellular microstructure, on the one hand, affects the transient temperature distribution resulting from an instantaneous source and temporarily creates domains with temperatures as high as two times the applied temperature difference in the external boundaries, while on the other hand, blocks harmonic thermal excitations in a range of frequencies (thermal band gaps) from passing the material. Moreover, the relative density of cellular architectures significantly affects the characteristics of thermal band gaps; for example, the first thermal band gap width may be reduced by 80% when relative density of cellular architectures changes from 0.2 to 0.6. Our findings can pave a path towards developing architected multifunctional metamaterials with programmable thermal isolation/guiding properties.

Evanescent waves in a metabeam attached with lossy acoustic black hole pillars

In this paper, we suggest a metabeam having periodically distributed integrated acoustic black holes (IABHs). Compared to conventional embedded ABHs, the thickness of the base beam has not been modified, so it adapts well to application occasions with load-bearing requirements. The ABH pillars behave as dynamic vibration absorbers (DVAs) that not only provide abundant modes with high damping performance but also generate local resonance and Bragg scattering bandgaps to stop wave propagation. To characterize the evanescent waves, a wave and Rayleigh–Ritz method (WRRM) is proposed, based on the nullspace method (NSM) that assumes the response is constituted by the linear combination of the nullspace basis of the constraint matrix. The proposed WRRM is validated using numerical partial differential solutions. Analysis of the complex dispersion curves shows that multiple local resonance and Bragg scattering bandgaps take place in the whole frequency range of interest, and the damping effect can merge these bandgaps so as to form very wide attenuation bands. Not only that, remarkable evanescent waves are also observed in the passbands. The lattice constant and ABH length are studied parametrically. Results show that the larger the lattice constant, the more attenuation bands appear, and the longer ABH length can enhance the attenuation capability. Finally, the transmissibility and mean square velocity for a finite metabeam with 5 cells are investigated, indicating substantial wave damping in a broadband frequency range.

Ultrawide attenuation bands in gradient metabeams with acoustic black hole pillars

In this letter, it is suggested to attach acoustic black holes (ABHs) on a base beam with gradient parameters for wave attenuation in ultrawide bands. Three types of metabeams have been designed and compared. Analysis of the complex dispersion curves obtained by the wave and Rayleigh–Ritz method indicates that modifying the pillar length may cause some failure bands, which can be circumvented by changing the lattice constant. The forced vibrations of finite beams are recovered using the nullspace method, which has confirmed the merits of the metabeam with gradient lattice constant. This paper enriched the application range of ABHs.

A wave and Rayleigh–Ritz method to compute complex dispersion curves in periodic lossy acoustic black holes

In this work we first propose a wave and Rayleigh–Ritz method (WRRM) to analyze the vibrations of periodic systems. It is well-known that one of the main difficulties of the Rayleigh–Ritz method (RRM) is to find suitable basis functions that satisfy the problem boundary conditions. In the WRRM, this is done by expanding the system response as a superposition of the basis of the nullspace of the matrix defining the system periodic conditions. Therefore, the WRRM widens the potential of the RRM and constitutes a computationally efficient alternative to the wave and finite element method (WFEM) for structures requiring very fine meshes. Although the method is of general application, in this paper it is presented for an infinite periodic beam consisting of acoustic black holes (ABHs) cells with damping layers. Periodic ABHs exhibit broadband vibration reduction thanks to bandgap formation at low frequencies and to the ABH effect in the mid-high frequency range. The WRRM allows one to obtain the equations of motion of the periodic ABH beam and to define linear and quadratic eigenvalue problems to respectively obtain the system’s real and complex dispersion curves. While the former have been studied at extent, the latter have been barely analyzed for periodic ABH beams or plates. And they are of critical importance given that ABHs are strongly damped systems and that damping layer properties may be frequency-dependent. Therefore, the second contribution of this work is to carefully inspect the role played by the imaginary part of the complex dispersion curves on the functioning of the ABH periodic beam. Its value below and above the ABH cut-on frequency is inspected and a parametric analysis considering variations on the ABH profile and damping layer geometry is carried out. The underlying physics is explained and, finally, a transmissibility analysis on a finite ABH periodic beam with five cells is provided to confirm the mechanisms found for infinite periodic damped ABH beams.

Thermal wave crystals based on the dual-phase-lag model

Thermal wave crystals based on the dual-phase-lag model are investigated in this paper by both theoretical analysis and numerical simulation to control the non-Fourier heat conduction process. The transfer matrix method is used to calculate the complex dispersion curves. The temperature response is calculated by the finite difference time domain method. The results show that thermal band-gaps exist due to the Bragg-scattering. The thermal wave impedance and center frequencies are introduced to predict band-gaps theoretically. The key parameters characterizing the band-gaps are analyzed. Our results show that the larger the difference in the thermal wave impedances is, the wider of the thermal band-gaps will be. This study demonstrates a type of the thermal metamaterials which have potential innovative applications such as thermal imagining, thermal diodes and thermal waveguides for energy transmission.

Parametric Analysis of Complex Dispersion Curves for Flexural Lamb Waves in Layered Plates in the Low-Frequency Range

A combined asymptotical and iteration method is used to study dispersion curves for the case of dynamic bending of isotropically layered plates. Based on the explicit limit formulation of dispersion equation, asymptotics of roots are derived in closed form for large values of root moduli. The influence of elastic and geometric parameters of layers are analyzed. The existence of critical values of geometric parameters that correspond to change of the type of asymptotics is demonstrated. The errors of asymptotics are estimated, and an iterative method is proposed for calculating the exact values of roots in statics. A low-frequency long-wave asymptotics of complex dispersion curves is derived; its accuracy is the higher the lower the frequency and the greater the number of the curve are. It is also proved that each complex curve has a long flat segment, the length of which increases simultaneously with the number of curve. The dispersion curves themselves are also calculated by another specific iterative procedure. The fundamental bending mode is analyzed together with its purely imaginary sister. The existence of the additional purely imaginary curve at low frequency is proved. Examples of calculating the static roots and the dispersion curves for subcritical and supercritical values of geometrical parameters are presented, and the efficiency of the algorithm is estimated.

A Bloch wave reduction scheme for ultrafast band diagram and dynamic response computation in periodic structures

A variety of Reduced-Order Modelling (ROM) techniques have been developed for the Wave/Finite Element (WFE) Framework. However most of these techniques are not compatible with dynamic response computation or frequency-dependent problems. This paper introduces a new reduction strategy for the WFE method, enabling the computation of both the forced response and the complex dispersion curves of periodic structures modelled using large-sized finite element (FE) models. The method exploits the duality between Inverse and Direct Bloch formulations to build a reduced solution subspace, accounting for both propagating and evanescent behaviours, while ensuring high reduction factors. This reduction strategy therefore enables the resolution of a wider range of problems, including near/far field response computation in finite waveguides subjected to dynamic loadings, or vibroacoustic transmission/reflection problems. First, the method is used to compute dispersion curves and forced response in a duct. Then a large bi-stiffened structure is studied to evaluate the method's performances. The high frequency resolution provided by the proposed ROM allows us to explore a variety of propagation and guided resonances localization effects, hardly accessible otherwise. Furthermore, the considerable reduction factors enable fast wave dispersion analyses in large-scaled periodic structures, complex phononic crystals designs or locally resonant metamaterials.

Heat reduction by thermal wave crystals

Non-Fourier heat conduction models assume wave-like behavior does exist in the heat conduction process. Based on this wave-like behavior, thermal conduction controlled in a one-dimensional periodical structure, named thermal wave crystal, has been demonstrated through both theoretical analysis and numerical simulation based on the Cattaneo-Vernotte (CV) heat-conduction model. The transfer matrix method and Bloch theorem have been applied to calculate the complex dispersion curves of thermal wave propagating in thermal wave crystals. And the temperature responses are obtained by using the FDTD method. The results show that the band gaps with pronounced heat reduction do exist in non-Fourier thermal transfer process because of the Bragg scattering. The mid-gap frequency is well predicted analytically based on the Bragg scattering mechanism. Finally, the key parameters determining the band gaps are presented and discussed. This study shows the potential applications of these materials in heat isolation and reduction.

Complex dispersion relations and evanescent waves in periodic beams via the extended differential quadrature method

Because of computation difficulties, investigations about the complex band structures and the evanescent wave modes in phononic crystals are very limited. In this paper, a novel k(ω) method, referred to as the Extended Differential Quadrature Element Method (EDQEM), is successfully developed to investigate the complex dispersion relations and the evanescent wave modes in periodic beams. At first, based on the Bloch-Floquet theorem and the two widely used beam theories, i.e., the Euler-Bernoulli beam theory and the Timoshenko beam theory, the EDQEM is developed to solve the dispersion equations of flexural waves in periodic beams. Comparisons with other related investigations are conducted to validate the correctness of the proposed method. Furthermore, considering three important factors, the shape of the unit cell, the pattern of the sampling point as well as the number of the sampling point, the convergence of the proposed method is investigated. Second, with the help of the EDQEM, complex dispersion relations of periodic beams are investigated and wave mode analysis is conducted, from which all possible waves, including propagative waves, purely evanescent waves and complex waves, in the complex dispersion curves are discussed. It is found that complex wave modes in periodic beams arise from two situations: (1) at the boundary of the first Brillouin zone and (2) within the first Brillouin zone. These complex wave modes are the transition modes between two propagative waves, or between the propagative wave and the complex wave. When the damping effect is included, all waves in periodic beams transfer into the complex waves. And, band gaps are not truly apparent anymore.

Low-frequency asymptotics of complex dispersion curves for lamb waves in layered elastic plate

An asymptotic and iterative method is proposed to calculate complex dispersion curves for isotropically layered plates. At the first stage, a dispersion equation is derived in explicit form and its limiting form is obtained for the static case. Passages to the limit of coinciding materials or vanishingly small layer thicknesses are investigated. Specific case of materials with coinciding shear moduli is analyzed in detail. Asymptotics of static roots is deduced for a large value of the root magnitude, the error of asymptotics is estimated, and an iterative method is proposed for calculating exact root values. Long-wave asymptotics of dispersion curves is derived, and it is shown that every complex dispersion curve has a long flat initial segment. Asymptotics is the more accurate, the lower the frequency is and the higher the number of the curve is. Exact values of wave numbers on the dispersion curve are also evaluated by another iterative procedure. Examples of calculating the dispersion curves are presented and the efficiency of the algorithm is shown.

On the impact of damping on the dispersion curves of a locally resonant metamaterial: Modelling and experimental validation

Recently, locally resonant metamaterials have come to the fore in noise and vibration control engineering, showing great potential due to their superior noise and vibration attenuation performance in targeted and tunable frequency ranges, referred to as stop bands. Damping has an important influence on the performance of these materials, broadening the frequency range of attenuation at the expense of peak attenuation. As a result, understanding and including the effects of damping is necessary to more accurately predict the attenuation performance of these locally resonant metamaterials. Classically, these often periodic structures are analysed using a unit cell modelling approach to predict wave propagation and thus stop band behaviour, discarding damping. In this work, a unit cell method including damping is used to analyse the complex dispersion curves of a locally resonant metamaterial design. The wave solutions in and around the stop band frequency range are compared to those of the unit cell method discarding damping. The influence on the dispersion curves of damping in resonator and host structure is discussed and the obtained dispersion curves are validated through an experimental dispersion curve measurement based on an Extended Inhomogeneous Wave Correlation method using Scanning Laser Doppler Vibrometry for a metamaterial plate manufactured at a representative scale. Excellent agreement is obtained between the numerically predicted and experimentally retrieved dispersion curves.

Light harvesting with metasurfaces: applications to sensors and energy generation

Metasurfaces have been receiving increasing interest due to the complex array of radiation controlling properties that are possible with single-layer films. This talk and work will focus on metasurfaces that can provide light filtering according to wavelength, polarization, and other properties of an incident beam, and applied to a variety of sensors. Also, metasurfaces that exhibit light trapping and localization that can be used for energy generation will be described. One last group of metasurfaces will be discussed that display complex dispersion curves that exhibit fast- and slow-light properties that can be used to study the complex electromagnetic phenomena.

Mode bifurcation and fold points of complex dispersion curves for the negative index metamaterial Goubau line

In this paper the complex dispersion curves of the four lowest-order transverse magnetic modes of a dielectric Goubau line (ϵ>0,μ>0) are compared with those of a weakly dispersive negative index metamaterial (NIM) Goubau line. The vastly different dispersion curve structure for the NIM Goubau line is characterized by unusual features such as mode bifurcation, complex fold points, both proper and improper complex modes, and merging of complex and real modes.

Resistivity and thermopower calculations in half-HeuslerTi1−xScxNiSn alloys from the KKR-CPA method

The Korringa–Kohn–Rostoker method with the coherent potential approximation (KKR-CPA)was applied to calculate electronic structure and transport coefficients in disorderedTi1−xScxNiSn, where semiconductor-to-metal crossovers have recently been observed in resistivity andthermopower experiments (Horyn et al 2004 J. Alloys Compounds 363 10; Romaka et al 2005J. Alloys Compounds 396 64). We have investigated the effect of chemical disorder onFermi surface features (i.e. group velocities and life-times related to real and imaginaryparts of complex dispersion curves, respectively). This analysis allowed us to shedlight on the principal mechanism responsible for the variation of thermoelectricproperties. Using the well-known formulae and performing integration over thecomplex Fermi surface, the residual conductivity and the thermopower slopeS/T were estimated. Satisfying agreement between theoretical results and experimental data (measuredat 90 and 300 K) has been found. This may indicate that modifications of electronic structure nearEF are predominantly responsible for the strong decrease of thermopower inTi1−xScxNiSn.