In-plane Elastic Waves Research Articles

Evanescent waves in two-dimensional fluid-saturated porous metamaterials with a transversely isotropic matrix

Wave propagation in a two-dimensional periodic fluid-saturated porous metamaterial (FSPM) is investigated. The constitutive relation considered for fluid-saturated porous materials is based on Biot’s homogeneization theory. Such media generally support two shear and two longitudinal elastic waves. Anisotropic wave propagation results both from anisotropy of the solid matrix and from the periodic structure of the metamaterial. Special attention is devoted to the dispersion and attenuation of evanescent Bloch waves in a FSPM whose matrix is transversely isotropic in which case in-plane elastic waves are a superposition of one shear wave and of two longitudinal waves. Bloch waves, complex band structures, and transmission properties are obtained numerically using finite element analysis. The case of homogeneous fluid-saturated porous media is considered since numerical simulations can be compared directly with analytic results in this case. The effects of material anisotropy and of fluid viscosity on wave propagation in two-dimensional FSPM are then discussed. It is found that wave-number band gaps appear in the complex band structure of the lossless FSPM due to the interference of waves with different but nonorthogonal polarizations. Wave-number band gaps are connected continuously by complex-frequency bands, hence, defining stop bands in the time domain. No complete frequency band gaps are found in two-dimensional lossless FSPM, in contrast to the one-dimensional case [Y.-F. Wang et al. Phys. Rev. B 99, 134304 (2019)] due to the presence of the additional quasishear wave accompanying the two quasi-longitudinal waves. Both the complex band structure and the transmission properties are affected by the anisotropy of the solid matrix. Wide transmission dips come up when viscosity is introduced as a result of the strong attenuation and the coupling of all wave polarizations. Concurrently, wave-number band gaps are washed out by viscosity. This theoretical paper has relevance to practical applications of fluid-saturated porous metamaterials, e.g., in concrete structures and geological soils.

Scattering of in-plane elastic waves at metamaterial interfaces

In this paper, we consider the problem of the scattering of in-plane waves at an interface between a homogeneous medium and a metamaterial. The relevant eigenmodes in the two regions are calculated by solving a recently described non self-adjoint eigenvalue problem particularly suited to scattering studies. The method efficiently produces all propagating and evanescent modes consistent with the application of Snell’s law and is applicable to very general scattering problems. In a model composite, we elucidate the emergence of a rich spectrum of eigenvalue degeneracies. These degeneracies appear in both the complex and real domains of the wave-vector. However, since this problem is non self-adjoint, these degeneracies generally represent a coalescing of both the eigenvalues and eigenvectors (exceptional points). Through explicit calculations of Poynting vector, we point out an intriguing phenomenon: there always appears to be an abrupt change in the sign of the refraction angle of the wave on two sides of an exceptional point. Furthermore, the presence of these degeneracies, in some cases, hints at fast changes in the scattered field as the incident angle is changed by small amounts. We calculate these scattered fields through a novel application of the Betti–Rayleigh reciprocity theorem. We present several numerical examples showing a rich scattering spectrum. In one particularly intriguing example, we point out wave behavior which may be related to the phenomenon of resonance trapping. We also show that there exists a deep connection between energy flux conservation and the biorthogonality relationship of the non self-adjoint problem. The proof applies to the general class of scattering problems involving elastic waves (under self-adjoint or non self-adjoint operators).

Cloaking In-Plane Elastic Waves with Swiss Rolls.

We propose a design of cylindrical cloak for coupled in-plane shear waves consisting of concentric layers of sub-wavelength resonant stress-free inclusions shaped as Swiss rolls. The scaling factor between inclusions’ sizes is according to Pendry’s transform. Unlike the hitherto known situations, the present geometric transform starts from a Willis medium and further assumes that displacement fields in original medium and in transformed medium remain unaffected (). This breaks the minor symmetries of the rank-4 and rank-3 tensors in the Willis equation that describe the transformed effective medium. We achieve some cloaking for a shear polarized source at specific, resonant sub-wavelength, frequencies, when it is located in close proximity to a clamped obstacle surrounded by the structured cloak. The structured medium approximating the effective medium allows for strong Willis coupling, notwithstanding potential chiral elastic effects, and thus mitigates roles of Willis and Cosserat media in the achieved elastodynamic cloaking.

高山-峡谷复合地形对入射平面P-SV波的散射

As a common composite site, the hill-canyon topography has complex coherence effects on seismic wave scattering. However, the publications of related research are still very limited. Therefore, a multi-domain indirect boundary element method was proposed to solve the scattering of in-plane elastic waves, and the scattering and diffraction of plane P-SV waves by hill-canyon topography were studied. The method takes advantages of both full-space Green’s function and half-space Green’s function to construct the scattered field in the independent closed region and the half-space open region, respectively. Combined with the auxiliary function method, dynamic wavefield solutions for the hill-canyon site were given, with the calculation accuracy ensured and the solution efficiency improved. The proposed method was verified through comparison of the results with published ones, and numerical calculations were performed in the frequency domain and time domain in the case of a Gaussian hill-canyon site. Results show that, the distribution of the surface displacement amplitude is very complex. There is significant dynamic interaction between the hill and the canyon, and the frequency-domain response depends on the frequency and angle of the incident wave. The presence of hills has an inhibition effect on the canyon for vertically incident waves, which significantly changes the peak acceleration and spectrum characteristics inside the canyon. Different height-to-width ratios of hills will cause changes in the seismic effect, and the presence of bedrock will also obviously amplify the seismic effect of the overall terrain.

High impact resistance in graphyne.

Graphyne was recently facilely synthesized with superior mechanical and electrical performance. We investigate the ballistic protection properties of α-, β-, δ-, and γ-graphyne sheets using molecular dynamics simulations in conjunction with elastic theory. The velocities of the in-plane elastic wave and out-of-plane cone wave are obtained by both membrane theory and molecular dynamics simulations. The specific penetration energies are approximately 83% that of graphene, indicating high impact resistance. γ-Graphyne has high sound wave speeds comparable to those of graphene, and its Young's modulus is approximately 60% that of graphene. δ-Graphyne has the highest cone wave speed among the four structures, while α-graphyne possesses the highest penetration energy and impact resistance at most tested projectile speeds. Our results indicate that graphyne is a good protective structural material.

Nonreciprocal transmission for the SH wave in a nonlinear elastic wave metamaterial with imperfect interfaces

During the past decades, phononic crystals and elastic wave metamaterials as the artificial periodic structures have received considerable attention. Due to their distinguishing properties, these structures can be used for the wave energy manipulation, e.g., band gaps, elastic wave cloaking and topological state, etc. A well-known characteristic is the band gap which means that the elastic wave cannot propagate in some certain frequency regions. Based on this property, many engineering applications can be achieved, e.g., filtering and wave isolation, etc. The investigations mentioned above are mainly focused on the linear elastic problems which cannot present the nonlinear wave properties. However, the nonlinearity appearing in the materials and structures can bring new and interesting wave phenomena. One of the nonlinear characteristics is the generation of the higher order harmonics. It can be applied to the nonlinear nondestructive testing. Recently, increasing attention has been focused on the nonlinear wave motion, which can result in the nonlinear band gap and acoustic diode. The new concept named as the acoustic diode has been presented by the combination of the band gap and material nonlinearity, which permits the wave propagation in only one direction. It can be achieved when the fundamental wave sits in the stop band but the second harmonic belongs to the pass band. However, the acoustic wave has one direction displacement component and usually propagates in the air or liquid. Thus we wonder whether the nonreciprocal phenomenon of the elastic wave can be achieved in the solid materials. The problem will become more interesting and necessary because of the coupling displacements with the vector characteristic in solids. In our previous work, we have discussed the nonreciprocal transmission for the incident SH and in-plane elastic waves in a nonlinear elastic wave metamaterial. The interface in the consecutive layers is always assumed as perfect, which means that the displacement and stress components are continuous across the interface. But perfect interfaces may become imperfect in practice during the manufacture, which indicates that the interfacial property on the band gaps is obvious. Then the imperfect interface can offer a new opportunity to tune the nonreciprocal transmission of the elastic wave which depends on the band gap property. In this work, the nonreciprocal transmission for the SH wave in a layered nonlinear elastic wave metamaterial with the imperfect interfaces is investigated. The combination of the structural asymmetry and material nonlinearity breaks the inherent reciprocity of the classic wave system. The second harmonic can be generated by the interaction between the incident SH wave and material nonlinearity. Based on the Bloch’s law and stiffness matrix method, the band gaps and transmission coefficients of both the fundamental and second harmonic waves are obtained. The effects of the interfacial properties on the nonlinear phononic crystal and elastic wave metamaterial are discussed. We find that comparing with the perfect interface system, the central frequency of the nonreciprocal regions shifts towards the low frequency region by the effects of imperfect interfaces. This present work is expected to be helpful to design the practical devices with the tuning nonreciprocal transmission of the elastic wave.

Elastic wave propagation in hierarchical lattices with convex and concave hexagons stacked vertexes.

The paper investigates the in-plane elastic wave propagation in hierarchical lattices. The hierarchical organization is obtained by removing a certain amount of cells successively from a general hexagonal lattice that involves the re-entrant configuration with negative Poisson's ratio. The symmetry reduction caused by the alteration of the internal angle and the formed vertexes with stacking hexagons gives rise to a significant effect on the dispersion properties of the structure materials. Results show that the lattices with different hierarchy levels possess a stable behavior of the band gap in the position near the resonant frequency of the cell walls, and the gap width has an evident increase in the re-entrant hierarchical structures. In addition, band structure in the low frequency range has an unchanged performance for the varying parameters of cell walls, and the reason is owing to the steady of the structure vibration pattern for the Bloch modes. The authors' findings of the dispersion behaviors in the hierarchical lattices provide a broad design space for the lightweight materials in the wave manipulation and vibration control.

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

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

A Petrov-Galerkin finite element interface method for interface problems with Bloch-periodic boundary conditions and its application in phononic crystals

In this paper, we propose a Petrov-Galerkin finite element interface method (PGFEIM) to solve the elliptic and elastic interface problems with Bloch-periodic boundary conditions. The main idea of this method is to choose the standard finite element basis independent of the interface to be the test function basis, and choose a piecewise linear function satisfying the jump conditions across the interface to be the solution basis. The grid we use is a non-body-fitted grid. The PGFEIM is capable of dealing with sharp-edged interface problems with matrix coefficients and nonhomogeneous jump conditions. Further, we extend this method to compute the band structure of anti-plane transverse waves and in-plane elastic waves in phononic crystals. Different acoustic impedance ratios, arbitrary complex scatterer geometries and various material properties are considered and discussed. Numerical experiments demonstrate that the PGFEIM is nearly second order accurate in the L∞ norm for interface problems, and it is accurate and efficient for computing the band structure of phononic crystals.

A modified smoothed finite element method (M-SFEM) for analyzing the band gap in phononic crystals

The present work proposes a novel modified smoothed finite element method (M-SFEM) to calculate the band structures of two-dimensional in-plane elastic waves in phononic crystals (PCs). Using the gradient smoothing technique over the cell-based smoothing domains, the cell-based gradient smoothing operation can offer ‘proper softening effects’ in SFEM modeling. According to the generalized integration rules, by simply shifting integration points to an unconventional location in the mass matrix, the accuracy of the M-SFEM model can be further improved. Numerical examples are presented for PCs using the proposed method for the computation of band gap (BG) frequency regions. The accuracy and efficiency of the modified SFEM are compared with those of the corresponding FEM and SFEM. The advantages of the modified SFEM for computing the BGs in PCs are discussed as compared to the conventional FEM. The results show the performance improvement of the M-SFEM compared to the FEM and SFEM.

Phononic topological insulators based on six-petal holey silicon structures

Since the discovery of the Quantum Spin Hall Effect, electronic and photonic topological insulators have made substantial progress, but phononic topological insulators in solids have received relatively little attention due to challenges in realizing topological states without spin-like degrees of freedom and with transverse phonon polarizations. Here we present a holey silicon-based topological insulator design, in which simple geometric control enables topologically protected in-plane elastic wave propagation up to GHz ranges with a submicron periodicity. By integrating a hexagonal lattice of six small holes with one central large hole and by creating a hexagonal lattice by themselves, our design induces zone folding to form a double Dirac cone. Based on the hole dimensions, breaking the discrete translational symmetry allows the six-petal holey silicon to achieve the topological phase transition, yielding two topologically distinct phononic crystals. Our numerical simulations confirm inverted band structures and demonstrate backscattering-immune elastic wave transmissions through defects including a cavity, a disorder, and sharp bends. Our design also offers robustness against geometric errors and potential fabrication issues, which shows up to 90% transmission of elastic waves even with 6% under-sized or 11% over-sized holes. These findings provide a detailed understanding of the relationship between geometry and topological properties and pave the way for developing future phononic circuits.

A design of active elastic metamaterials with negative mass density and tunable bulk modulus

In this paper, a design of active elastic metamaterial that possesses negative density and tunable bulk modulus is presented for the negative refraction of in-plane elastic waves at deep subwavelength scale. The metamaterial is fabricated in an aluminum plate, and the resonant structure in a unit cell of the metamaterial composes of a coated steel that functions as a translational resonator and a radially polarized piezoelectric transducer shunted with negative capacitance. Based on effective continuum theory, the effective mass density and bulk modulus are numerically determined. The passive and active elements work together to generate broad band double-negative material properties. Simulation results verified and demonstrated the negative refraction of elastic waves at the interface between proposed elastic metamaterials and natural solids. The proposed elastic metamaterial may thus be used as a flat lens with broad working frequency regime for in-plane elastic wave focusing.

Multiscale nonlocal effective medium model for in-plane elastic wave dispersion and attenuation in periodic composites

This manuscript proposes a multiscale nonlocal homogenization and a nonlocal effective medium model for in-plane wave propagation in periodic composites accounting for dispersion and attenuation due to Bragg scattering. The nonlocal effective medium model is developed based on the spatial-temporal nonlocal homogenization model that is formulated to capture dispersion within the first Brillouin zone with particularly high accuracy along high symmetry directions. The homogenization model is derived by employing high order asymptotic expansions, extending the applicability of asymptotic homogenization to short wavelength regime, to capture wave dispersion and attenuation. The effective medium model is in the form of a second order PDE with a nonlocal effective moduli tensor that contains the nonlocal features of the homogenization model. The proposed models are derived and numerically verified for in-plane elastic wave propagation in two-dimensional periodic composites. It is shown that the dispersion curves of the spatial-temporal nonlocal homogenization model capture the acoustic branch, the first stop band and the optical branch of longitudinal and shear wave modes. The nonlocal effective medium model predicts transient elastic wave propagation in periodic composites and captures wave dispersion and attenuation within the limits of separation of scales.

Topological phase transition in mechanical honeycomb lattice

Topological materials provide a new tool to direct wave energy with unprecedented precision and robustness. Three elastic topological phases, the valley Hall, Chern and spin Hall insulators, are currently studied, and they are achieved separately in rather distinct configurations. Here, we explore analytically various topological phase transitions for in-plane elastic wave in a unified mass-spring honeycomb lattice. It is demonstrated that the three elastic topological phases can be realized in this single lattice by designing mass, stiffness or introducing Coriolis’ effect. In particular, the interface between valley Hall and Chern insulators is found to support topological interface mode for the first time. Perturbation method is used to derive the analytic effective continuum model in the neighbor of band degeneracy, and the physics in topological phase transitions are revealed through evaluation of topological invariants. The topologically protected interface states, their decaying profile as well as the pseudo-spin-indicating polarization specific for elastic wave are systematically analyzed, and these results are further confirmed numerically by Bloch wave analysis of domain wall strip and transient simulation of finite sized sample. This study offers a concise and unified analytical model to explore topology nature of elastic wave, and can provide intuitive guidance to design of continuum mechanical topological materials.

In-plane elastic wave propagation in nanoscale periodic layered piezoelectric structures

Wave propagation in nanoscale piezoelectric structures is much more complex than in macro-sized elastic structures. This study is devoted to the analysis of in-plane elastic waves propagating normally and obliquely in nanoscale periodic layered piezoelectric structures. The equations of wave motion which take both the piezoelectric effect and the nanoscale size-effect into account are solved numerically. The transfer matrix method based on the nonlocal piezoelectric continuum theory is used to calculate the localization factor which characterizes the band structures. Additionally, the wave transmission spectra are also computed by the stiffness matrix method. The piezoelectric effect (i.e., the piezoelectric constants) and the size-effect (i.e., R, the influences of the ratio of the internal to external characteristic lengths) on the cut-off frequency and the wave propagation characteristics are discussed in details. The influence of the mode conversion on band-gaps and the influence of R on the mode conversion are analyzed.

Topologically protected edge states for out-of-plane and in-plane bulk elastic waves

Topological phononic insulators (TPnIs) show promise for application in the manipulation of acoustic waves for the design of low-loss transmission and perfectly integrated communication devices. Since solid phononic crystals exist as a transverse polarization mode and a mixed longitudinal-transverse polarization mode, the realization of topological edge states for both out-of-plane and in-plane bulk elastic waves is desirable to enhance the controllability of the edge waves in solid systems. In this paper, a two-dimensional (2D) solid/solid hexagonal-latticed phononic system that simultaneously supports the topologically protected edge states for out-of-plane and in-plane bulk elastic waves is investigated. Firstly, two pairs of two-fold Dirac cones, respectively corresponding to the out-of-plane and in-plane waves, are obtained at the same frequency by tuning the crystal parameters. Then, a strategy of zone folding is invoked to form double Dirac cones. By shrinking and expanding the steel scatterer, the lattice symmetry is broken, and band inversions induced, giving rise to an intriguing topological phase transition. Finally, the topologically protected edge states for both out-of-plane and in-plane bulk elastic waves, which can be simultaneously located at the frequency range from 1.223 to 1.251 MHz, are numerically observed. Robust pseudospin-dependent elastic edge wave propagation along arbitrary paths is further demonstrated. Our results will significantly broaden its practical application in the engineering field.

Scattering of In-plane Elastic Waves by an Anisotropic Circle

The 2D scattering of in-plane elastic waves by a circle is considered when the surrounding medium is isotropic and the medium inside the circle is anisotropic (orthotropic). The equations inside the circle are transformed to polar coordinates and then depend explicitly on the azimuthal angle through trigonometric functions. Making expansions in trigonometric series in the azimuthal coordinate give a coupled system of ordinary differential equations in the radial coordinate that is solved by power series expansions. With the solution inside the circle complete the scattering problem is solved essentially as in the classical case. The elements of the transition (T) matrix of the circle are given explicitly for low frequencies (long wavelengths). For low frequencies some numerical examples are given showing the strong influence of anisotropy.

In-plane elastic wave propagation in aluminum honeycomb cores fabricated by bonding corrugated sheets

Aluminum honeycomb cores are widely used in sandwich structures due to their high stiffness-to-weight ratios and very low densities. However, owing to their porous architecture, honeycomb cores are inherently week and are susceptible to damage due to inadvertent or improper loadings on their encompassing sandwich structure. This damage can potentially lead to the failure of the sandwich structure, and therefore it should be detected and evaluated, preferably using nondestructive methods. Common nondestructive techniques have limited effectiveness in inspecting aluminum honeycombs due to their porous structure and dispersive properties. Since honeycombs are less dispersive at sub-ultrasound frequencies, inspecting them using low and sub-ultrasound frequencies has been introduced lately as a promising alternative to ultrasound inspection. However, this approach requires a priori knowledge of the wave propagation characteristics in the inspected material, which is not readily available for most commercially available aluminum honeycombs, especially the ones manufactured by joining thin corrugated sheets. Thus, this work utilizes finite element computations to assess the low frequency wave propagation characteristics (i.e. phase velocity and dispersive properties) in commercially available aluminum honeycombs made by bonding thin corrugated sheets. Results illustrate that the dispersive behavior and acoustic anisotropy of the studied honeycombs are more significant at higher porosities and high frequencies as well as identify the frequencies below which honeycombs exhibit their least dispersive acoustic behavior.

A comparative study of wave localization in locally resonant Thue–Morse, Rudin–Shapiro and Period-Doubling aperiodic structures

The localization characteristics of the in-plane elastic waves in locally resonant aperiodic phononic crystals are examined in this study. In particular, the phononic crystals generated according to the Thue–Morse, Rudin–Shapiro and Period-Doubling sequences are theoretically investigated by using the extended transfer matrix method. For comparison, the binary and ternary locally resonant systems are considered, and their band structures are characterized by using the localization factors. Moreover, the influences of structural arrangement, material combination, incidence angle, number of components, length ratio, and random disorder on the band structures are also discussed. Some novel and interesting phenomena are observed and discussed.

Optical Measurement of In-plane Waves in Mechanical Metamaterials Through Digital Image Correlation

We report on a Digital Image Correlation-based technique for the detection of in-plane elastic waves propagating in structural lattices. The experimental characterization of wave motion in lattice structures is currently of great interest due its relevance to the design of novel mechanical metamaterials with unique/unusual properties such as strongly directional behaviour, negative refractive indexes and topologically protected wave motion. Assessment of these functionalities often requires the detection of highly spatially resolved in-plane wavefields, which for reticulated or porous structural assemblies is an open challenge. A Digital Image Correlation approach is implemented that tracks small displacements of the lattice nodes by centring image subsets about the lattice intersections. A high speed camera records the motion of the points by properly interleaving subse- quent frames thus artificially enhancing the available sampling rate. This, along with an imaging stitching procedure, enables the capturing of a field of view that is sufficiently large for subsequent processing. The transient response is recorded in the form of the full wavefields, which are processed to unveil features of wave motion in a hexagonal lattice. Time snapshots and frequency contours in the spatial Fourier domain are compared with numerical predictions to illustrate the accuracy of the recorded wavefields.