Phononic Plate Research Articles

Attenuation of Lamb waves in coupled-resonator viscoelastic waveguide

Guidance of elastic waves is one of the main applications of artificial crystal structures. The attenuation of the guided waves is, however, often overlooked, as most of the proposed waveguides only comprise ideal elastic materials. In this work, we study the propagation of evanescent Lamb waves guided in coupled-resonator viscoelastic waveguide (CRVW), with special attention to attenuation. CRVW is defined by considering a linear chain of coupled defect cavities in a phononic plate made of epoxy. The viscoelastic behavior of epoxy is characterized numerically by the Kelvin–Voigt (K–V) model. Based on finite element analysis, the complex band structure and the spectrum of frequency response function (FRF) are obtained. Due to viscosity, guided Lamb waves are spatially damped. Two theoretical models are devised to predict the displacement distributions inside and outside a bandgap for guided waves, respectively, considering either the first or the first two least evanescent Bloch waves identified in the complex band structure. A CRVW sample is fabricated and characterized experimentally by laser vibrometry. Evanescent Lamb waves are observed to be strongly confined along the waveguide and at the same time to decay rapidly along the waveguide axis. Experiments and numerical simulations are found to be in fair agreement. The present work is expected to inspire practical applications of highly confined viscoelastic phononic waveguides.

Maximizing band gaps of single-phase phononic plates: Isogeometric optimal approach and 3D printing experimental validation

This work presents an effective isogeometric shape optimization approach for finding and widening the phononic band gaps of single-phase Mindlin plate structures. As the single-phase material phononic plate is easy to be manufactured by additive manufacturing, it has been investigated here by optimized its thickness profile to find and widen the band gaps. The proposed method utilizes a coarse B-spline surface to model the thickness profile of periodic plate and a fine B-spline surface to model the mid-surface of plate structure for simulation. The optimal design variables are the thickness variables, i.e., the z-components of the control points of the coarse B-spline surface. To avoid specifying the initial control point locations manually, the constrained dynamic maximization problem is solved by a particle swarm optimization (PSO) algorithm here. Various numerical examples demonstrate the effectiveness and reliability of the proposed method in finding optimal phononic band gaps of periodic plates. And the numerical results show that there is no band gap for hmax = 3 hmin, and the band gaps can be found and widen for hmax ≥ 5 hmin. The obtained band range is 638.9–823.6 Hz with a decreased central frequency of 731.25 Hz for hmax = 5 hmin and the width and the number of band gaps are increased as the maximum allowable thickness increases. Finally, one optimized design is fabricated through additive manufacturing, and the experimental frequency response is consistent with the results based on isogeometric analysis.

Multi-band topological valley modes of flexural waves in micro-perforated phononic plates

To develop phononic topological insulators (PTIs) with time reversal invariance, phononic crystals (PCs) are elaborately configurated to ensure the lattice symmetry and meanwhile to open the complete band gap. Particularly, a multi-band PTI requires more complex architecture than its single-band counterpart, leading to extra difficulty in implementation and poor manufacturability of PCs. Here, we present a straightforward and generic design strategy, to realize multi-band PTIs in micro-perforated phononic plates. Rather than using complex surface-mounted or embedded scatterers, the novel phononic plate is micro-perforated with a type of fan-shaped split-ring resonators (SRRs), the multi-modal resonances of which under constrained lattice symmetry induce multiple Dirac degeneracies in flexural-wave dispersions at different frequency scales. The multi-band topological phase transitions are observed, and triple topological band gaps are caused when rotating micro-perforated slots properly. Experimental results agree well with numerical simulation results, both demonstrating triband valley edge-state propagation of flexural waves. In addition, we also prove that the developed valley PTIs enable the second-order corner modes of flexural waves within the corresponding topological band gaps. Parametric studies further reveal that the emergence of the valley-shaped Dirac degeneracy bands in the developed PTI is insensitive to the plate thickness and slots parameters, and more than three Dirac cones exist in flexural-wave dispersions. The developed micro-perforated plates with SRRs provides a convenient platform for enabling multi-band PTIs with various symmetries, facilitating applications of multi-frequency topological effects of elastic waves in a cost-effective way.

Extended plane wave expansion formulation for viscoelastic phononic thin plates

The extended plane wave expansion (EPWE) formulation is derived to obtain the complex band structure of flexural waves in viscoelastic thin phononic crystal plates considering the Kirchhoff–Love plate theory. The presented formulation yields the evanescent behavior of flexural waves in periodic thin plates considering viscoelastic effects. The viscosity is modeled by the standard linear solid model (SLSM), typically used to closely model the behavior of polymers. It is observed that the viscoelasticity influences significantly both the propagating and evanescent Bloch modes. The highest wave attenuation of the viscoelastic phononic thin plate is found around a unit cell filling fraction of 0.37 for higher frequencies considering the least attenuated wave mode. This EPWE formulation broadens the suitable methods to handle evanescent flexural waves in 2-D thin periodic plate systems considering the effects of viscoelasticity on wave attenuation.

Topological vortex mode for flexural waves in pillared plates

Localized topological modes with high robustness to various perturbations are receiving increasing attention. Recently, zero-order topological vortex modes have been designed in phononic structures, in analogy with zero-energy fermionic states modulated by Jackiw-Rossi binding mechanism. Such localized modes may have potential applications for biosensing, bioimaging and on-chip communication. In this work, we propose a pillared phononic plate with a Kekulé distortion of pillars position to bind topological modes at a vortex core via dispersion engineering. The phase winding and amplitude diagrams of the topological vortex mode are observed experimentally. It is found that existence of vibration peaks and corresponding mode patterns are strongly robust against the random perturbation of resonant frequencies of pillars at the vortex core. We further design a topological resonant sensor for mass sensitivity. The frequency of the topological vortex mode is almost linearly sensitive to added small mass at the vortex core in terms of mass values and positions. The proposed pillared plates provide a platform for potential applications such as energy localization and harvesting, remote health monitoring.

Reconfigurable localized effects in non-Hermitian phononic plate

Skin effect is one of the intriguing phenomena exhibited by non-Hermitian wave systems. It reflects the localization of the modes at the boundaries of the structure. We demonstrated the skin effect for elastic waves propagating in a non-Hermitian phononic plate containing piezoelectric components in their unit cells. The latter behave as sensors and actuators by using the direct and inverse piezoelectric effects. The demonstration is based on the calculation of the complex non-reciprocal dispersion curves and their analysis for any direction of the wavevector in the two-dimensional space. Therefore, localization phenomena at different boundaries and corners of a finite square structure are presented. Furthermore, by applying different levels of non-Hermiticity in different parts of a square structure, it is shown that the localized features can appear at different positions and with various shapes. These localized phenomena can be reconfigured by acting on the non-Hermiticity parameters. Our results provided a feedback control strategy to introduce the non-Hermitian skin effect in two-dimensional elastic systems for potential applications, such as vibration control, energy harvesting, and sensing.

Deep-Learning-Based Acoustic Metamaterial Design for Attenuating Structure-Borne Noise in Auditory Frequency Bands

In engineering acoustics, the propagation of elastic flexural waves in plate and shell structures is a common transmission path of vibrations and structure-borne noises. Phononic metamaterials with a frequency band gap can effectively block elastic waves in certain frequency ranges, but often require a tedious trial-and-error design process. In recent years, deep neural networks (DNNs) have shown competence in solving various inverse problems. This study proposes a deep-learning-based workflow for phononic plate metamaterial design. The Mindlin plate formulation was used to expedite the forward calculations, and the neural network was trained for inverse design. We showed that, with only 360 sets of data for training and testing, the neural network attained a 2% error in achieving the target band gap, by optimizing five design parameters. The designed metamaterial plate showed a -1 dB/mm omnidirectional attenuation for flexural waves around 3 kHz.

Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate

This paper investigates the complex band diagram of flexural waves in the phononic plate with semi-infinite heavy fluid loading. The system under examination is a square plate lattice with two-dimensional periodicity immersed in a fluid domain with infinite height. The numerical models based on the wave field transformation and the Galerkin method combined with the finite element discretization technique are developed to investigate the real and imaginary parts of the dispersion relation of flexural waves propagating through the phononic plate incorporating the fluid-loading effects. A perfect agreement is found between the location and width of stop bands from the real band diagram and the attenuation diagram, which supports the validity of the numerical models. Moreover, the complex band diagram is verified by the transverse vibration transmittance of the finite phononic plate. The results demonstrate that the external fluid loading is able to adjust the location, bandwidth, and decaying level as well as affect the degree of attenuation anisotropy of the complete and directional band gaps.

Voltage-controlled topological interface states for bending waves in soft dielectric phononic crystal plates

The operating frequency range of passive topological phononic crystals is generally fixed and narrow, limiting their practical applications. To overcome this difficulty, here we design and investigate a one-dimensional soft dielectric phononic crystal (PC) plate system with actively tunable topological interface states via the mechanical and electric loads. We use nonlinear electroelasticity theory and linearized incremental theory to derive the governing equations. First we determine the nonlinear static response of the soft dielectric PC plate subjected to a combination of axial force and electric voltage. Then we study the motion of superimposed incremental bending waves. By adopting the Spectral Element Method, we obtain the dispersion relation for the infinite PC plate and the transmission coefficient for the finite PC plate waveguide. Numerical results show that the low-frequency topological interface state exists at the interface of the finite phononic plate waveguide with two topologically different elements. By simply adjusting the axial force or the electric voltage, an increase or decrease in the frequency of the topological interface state can be realized. Furthermore, applying the electric voltage separately on different elements of the PC plate waveguide is a flexible and smart method to tune the topological interface state in a wide range. These results provide guidance for designing soft smart wave devices with low-frequency tunable topological interface states.

An Algorithm for the Complete Solution of the Quartic Eigenvalue Problem

The quartic eigenvalue problem (λ 4 A +λ 3 B +λ 2 C +λ D + E ) x = 0 naturally arises in a plethora of applications, such as when solving the Orr–Sommerfeld equation in the stability analysis of the Poiseuille flow, in theoretical analysis and experimental design of locally resonant phononic plates, modeling a robot with electric motors in the joints, calibration of catadioptric vision system, or, for example, computation of the guided and leaky modes of a planar waveguide. This article proposes a new numerical method for the full solution (all eigenvalues and all left and right eigenvectors) that, starting with a suitable linearization, uses an initial, structure-preserving reduction designed to reveal and deflate a certain number of zero and infinite eigenvalues before the final linearization is forwarded to the QZ algorithm. The backward error in the reduction phase is bounded column wise in each coefficient matrix, which is advantageous if the coefficient matrices are graded. Numerical examples show that the proposed algorithm is capable of computing the eigenpairs with small residuals, and that it is competitive with the available state-of-the-art methods.

3D intra-cellular wave dynamics in a phononic plate with ultra-wide bandgap: attenuation, resonance and mode conversion

The intra-cellular wave dynamics of a water jetted phononic plate are experimentally investigated by means of high-resolution three-dimensional (3D) scanning laser Doppler vibrometry. The study is focused on the vibrational behavior around the ultra-wide bandgap of the plate (with a relative bandgap width of 0.89), as the critical frequency range of its phononic functionality. Broadband vibrational excitations are applied using a piezoelectric transducer and both in-plane and out-of-plane operational deflection shapes of the unit-cells are analyzed with respect to mode shapes calculated by finite element (FE) simulation. Attenuation and resonance of both symmetric and antisymmetric wave modes are validated, and it is shown that despite the absence of in-plane wave energy actuation, the symmetric modes are effectively excited in the phononic lattice, due to mode conversion from co-existing antisymmetric modes. Supported by FE modal analysis, this mode conversion observation is explained by the slight through-the-thickness asymmetry introduced during manufacturing of the phononic plate which leads to coupling of modes with different symmetry. The results confirm the potential of such detailed 3D inspection of phononic crystals (and in general acoustic metamaterials) in gaining full insight about their intracellular dynamics, which can also illuminate discrepancies with respect to idealized numerical models that might be due to manufacturing imperfections.

Topological bound states in elastic phononic plates induced by disclinations

Topological bound states in elastic phononic plates induced by disclinations

Negative refraction of flexural wave propagation on phononic thin plates based on the backward wave effect

The negatively refracted flexural wave is explored for phononic thin plates consisting of periodic holes and inclusions based on the effect of backward waves. Dispersion curves of these two types of phononic thin plates are calculated by using the multipole method, and the backward wave effect has been clearly found in their band diagrams. By using multiple scattering simulations, the birefraction phenomenon has been observed in a prism-shaped hole array, whereas single negative refraction has been realized in a prism-shaped inclusion array. The explanations using Snell’s law are presented for the occurrence of birefraction and sole negative refraction. Experiment verifications of birefraction phenomenon show good agreements with both numerical and analytical results. Numerical results also indicate the multiple scattering simulation is an alternative method for observing negative refraction of flexural waves.

Topologically valley-polarized edge states in elastic phononic plates yielded by lattice defects

Lattice defects (including disclinations and dislocations) which commonly exist in various materials, have shown a fantastic ability to produce excellent mechanical and physical properties of matters. In this paper, lattice defects are introduced into valley-polarized elastic phononic plates. Deformations of lattices yield interfaces expressing as topologically protected waveguides, due to the valley-polarized phase transition of phononic crystals (PnCs) across interfaces. The topological interface states immunize against the finite sizes and the moderate disturbances of plates, essentially differing from the trivial lattice defect states. The discovery of topologically valley-polarized edge states yielded by lattice defects unveils a new horizon in topological mechanics and physics. It provides a novel platform to implement elastic devices with robust topological waveguides.

Tunable multidispersive bands of inductive origin in piezoelectric phononic plates

A variety of multidispersive, localized, or extended in frequency, bands, induced by inductance-based external electric circuits in piezoelectric phononic plates, is studied both theoretically and experimentally in this work. Their origin, tightly related to an equivalent LC-circuit behavior, is analyzed in detail and their interaction with the Lamb-like guided modes of the plate is also discussed. These bands, easily tuned by the choice of the parameters of the external electric circuitry, lead to a non-destructive, real-time control of the dispersion characteristics of these structures. Our device and analysis can find application in the improvement of surface acoustic wave components by offering additional degrees of freedom.

Crosstalk reduction in a piezoelectric phononic plate induced by electrical boundary conditions: Application to multi-element transducers

This paper proposes a method of passive electrical decoupling which aimed at found application in reducing the crosstalk phenomenon in multi-element ultrasonic transducers. A homogeneous piezoelectric plate, covered on one side by a 1D periodic arrangement of thin metallic electrodes and on the other side by a full electrode, is considered. Finite element analysis and experimental measurements are performed to obtain the dispersion curves and normal displacements at the surface of the structure. It is shown that applying inductive shunts at the electrodes, band gaps can be created in the first Brillouin zone, which can prevent from the establishment of the first thickness mode in the plate. In that way the mechanical inter-element coupling can be lowered. Thus, the acoustic radiation in water from one sector excited at the resonance frequency is found to be closer to that of a piston mode. The transposition of this principle to the situation of a transducer including a rear medium and a front matching layer confirms the possibility of reducing the inter-element coupling. However, the physical effects at the origin of this reduction are different from those inherent to the cutting of a piezocomposite ceramic as it is done in most probes available in the market. As a result, we show that taking advantage of the electrical boundary conditions upon the passive elements in a transducer gives real opportunities for crosstalk reduction that may be implemented in ultrasonic systems for imaging in the medical field and in NDT.

A novel hybrid composite phononic crystal plate with multiple vibration band gaps at low frequencies

The low-frequency vibration isolation of a new hybrid phononic crystal plate are numerically and experimentally demonstrated in this paper. The proposed structure is composed of two types of periodic double-sided composite resonators deposited on a two-component plate. The dispersion relations, transmission spectra, and displacement fields of the eigenmodes are calculated by the finite element method. Compared to the classical phononic plate, the proposed structure can generate multiple flexural and longitudinal band gaps (BGs) at low frequencies. The formation mechanisms of the BGs can be explained by the corresponding “analogous-rigid modes”. The results of the parametric analyses show that the BGs can be significantly modulated by adjusting the material and geometric parameters. Furthermore, shake tests on a fabricated specimen were performed to verify the attenuation performance for elastic waves of specific frequency ranges. This research indicates that the proposed structure is a promising option for broadband low-frequency vibration insulation.

Topological cavities in phononic plates for robust energy harvesting

Piezoelectric energy harvesting has attracted tremendous interest for designing sustainable self-powered devices/systems targeted to special environment such as wireless or wearable applications. The traditional cavity (e.g., phononic cavity mode) excitation is highly applicable in terms of sufficient power generation, nevertheless, has to endure the drawback of extremely poor robustness intrinsic to the trivial cavity modes. We propose to use phononic thin plate systems for robust energy harvesting application relying on zero-dimensional cavities confined by the Kekulé distorted topological vortices. The harvesting power induced by topological cavities is about 30 times that of the bare plate. Further studies on the effects of deliberately introduced defects on the output power show that the proposed energy harvesting system is highly robust against symmetry-preserving defects, and is less influenced even for symmetry-breaking defects at moderate perturbation level. Beyond the reported energy harvesting application, we foresee that our work may open avenues for robust operations in the realm of wireless sensing and structural health monitoring.

Lamb waves propagating in one-dimensional phononic composite and nesting Fibonacci piezoelectric superlattices plate coated on a uniform substrate

In this paper, we present the dispersion relation of Lamb waves propagating along one-dimensional phononic and along nesting Fibonacci superlattices, using the plane wave expansion and stiffness matrix methods. More band gaps of Lamb wave appear in nesting Fibonacci superlattices plates than in phononic composite plates. Effect of substrate on band gap structures of nesting Fibonacci composite model is analyzed. In addition, the influence of widths of the materials’ strips on the band structure is discussed. Finds could be relevant to understand the intrinsic physical properties of quasi-periodic composites piezoelectric and provide flexible choices to meet real engineering applications.

Experimental full wavefield reconstruction and band diagram analysis in a single-phase phononic plate with internal resonators

Research on phononic crystal architectures has produced many interesting designs in the past years, with useful wave manipulation properties. However, not all of the proposed designs can lead to convenient realizations for practical applications, and only a limited number of them have actually been tested experimentally to verify numerical estimations and demonstrate their feasibility. In this work, we propose a combined numerical-experimental procedure to characterize the dynamic behavior of metamaterials, starting from a simplified 2D design to a real 3D manufactured structure. To do this, we consider a new design of a resonator-type geometry for a phononic crystal, and verify its wave filtering properties in wave propagation experiments. The proposed geometry exploits a circular distribution of cavities in a homogeneous material, leading to a central resonator surrounded by thin ligaments and an external matrix. Parametric simulations are performed to determine the optimal thickness of this design, leading to a large full band gap in the kHz range. Full-field experimental characterization of the resulting phononic crystal using a scanning laser Doppler vibrometer is then performed, showing excellent agreement with numerically predicted band gap properties and with their resulting effects on propagating waves. The outlined procedure can serve as a useful step towards a standardization of metamaterial development and validation procedures.