Anti-plane Waves Research Articles

A Semi‐Analytical Method for Simulating Near‐Field Antiplane Wave Propagation in Layered Fluid‐Saturated Porous Media

ABSTRACTA semi‐analytical method for the near‐field antiplane wave propagation analysis in the layered fluid‐saturated porous media (FSPM) is proposed based on the Biot u–U dynamic formulation. The wave propagation equations of the FSPM are decoupled by the variable‐separating method. The thin‐layer element method (TLEM) is applied to discretize the infinite domain and construct the consistent artificial boundary condition. The finite element method (FEM) is adopted for the space discretization of the finite domain and the numerical solution of the dynamic response. The proposed method is validated by the comparison of the numerical results of this method with those in the published references and acquired from the remote artificial boundary. Subsequently, this method is applied to investigate typical near‐field antiplane wave propagation problems in the FSPM. Parametric sensitivity investigations are also executed to explore the impact of mechanical parameters, including permeability coefficients, porosity, and shear modulus of the solid phase, on the dynamic response of the FSPM. The study results confirm the efficacy and efficiency of the proposed method in the near‐field antiplane wave propagation analysis in the FSPM.

Analytical and numerical study of anti-plane elastic wave scattering in a structured quadrant subjected to a boundary point load

We verify the recently developed method by Kisil (Kisil 2024 SIAM J. Appl. Math. 84 , 464–476 (doi: 10.1137/23M1562445 )) through its application in the study of the anti-plane dynamic response of an elastic square-cell lattice quadrant having a free lateral boundary that is subjected to a sinusoidal point load. The vertical boundary of the quadrant is assumed to be either fixed or free. The approach adopted utilizes the discrete Fourier transform in both principal lattice directions to reduce the governing equations to a functional equation linking both the bulk and boundary response of the system. For both problems, this functional equation is shown to reduce to scalar Wiener–Hopf equation whose solution can be exploited to determine the full response of the lattice system. The analytical solutions based on the Wiener–Hopf method are also implemented in numerical investigations that demonstrate the lattice response to various positions and frequencies of the load. In particular, this study includes dynamic regimes induced by the load that possess strong anisotropy and shows how the associated waveforms interact with the boundaries of the quadrant.

On effective surface elastic moduli for microstructured strongly anisotropic coatings

The determination of surface elastic moduli is discussed in the context of a recently proposed strongly anisotropic surface elasticity model. The aim of the model was to describe deformations of solids with thin elastic coatings associated with so-called hyperbolic metasurfaces. These metasurfaces can exhibit a quite unusual behaviour and concurrently a very promising wave propagation behaviour. In the model of strongly anisotropic surface elasticity, strain energy as a function of the first and second deformation gradients has been introduced in addition to the constitutive relations in the bulk. In order to obtain values of surface elastic moduli, we compare dispersion relations for anti-plane surface waves obtained using the two-dimensional (2D) model and three-dimensional (3D) straightforward calculations for microstructured coatings of finite thickness. We show that with derived effective surface moduli, the 2D model can correctly describe the wave propagation.

Metamaterial design enabling simultaneous manipulation of Rayleigh and Love waves

Studies on elastic metamaterials have expanded from manipulating bulk waves to surface waves, aiming to control the propagation of in-plane Rayleigh waves or anti-plane Love waves. Considering the coexistence of Rayleigh and Love waves in various scenarios, the objective of this study is to develop a metamaterial capable of simultaneously manipulating both types of waves. The proposed metamaterial consists of horizontal resonators with an oblique mounting angle relative to the wave propagation direction, as well as vertical resonators. Initially, analytical solutions for the dispersion of surface waves are derived, followed by Finite Element (FE) simulations to validate the analytically predicted dispersion and illustrate the corresponding wave modes, as well as the in-plane and out-of-plane displacement fields at specified frequencies. The present study reveals that the mounting angle of the horizontal resonators plays a crucial role in surface wave manipulation. By adjusting the mounting angle, three distinct objectives can be achieved: (i) the attenuation of Rayleigh waves alone; (ii) the independent attenuation of Rayleigh and Love waves, targeting different frequency ranges; and (iii) the simultaneous attenuation of both Rayleigh and Love waves.

Convolution quadrature time-domain boundary element method for antiplane anisotropic viscoelastic wave propagation

This paper presents a Convolution Quadrature Time-Domain Boundary Element Method (CQBEM) for antiplane anisotropic viscoelastodynamics. The proposed CQBEM formulation employs the standard linear viscoelastic model and the fundamental solution developed by Wang and Achenbach for expressing viscoelastic and anisotropic properties, respectively. This fundamental solution, which involves integration over the unit circle, is evaluated numerically. After offering a comprehensive overview of the proposed CQBEM formulation, we demonstrate the effectiveness of the proposed method by analyzing the scattering of elastic waves by cavities in certain types of anisotropic viscoelastic materials. Additionally, the use of group velocity curves further validates the numerical results, confirming the anisotropic and viscoelastic effects and thus substantiates the proposed CQBEM for antiplane anisotropic viscoelastic wave propagation.

Anti-plane waves in a liquid-loaded piezo-flexo-electric layered model with interface energy

The basic design of a Love wave (LW) bio-sensor contains the loading of a viscoelastic liquid on top of a layered structure with distinct viscoelastic properties. Changes in the characteristics of the propagating acoustic wave caused by biochemical interactions at the sensing area can be detected at the output Inter digital transducer (IDT). The propagation of the anti-plane (AP) wave is discussed in this work in a layered structure of piezo-flexo-electric (PFE) layer bonded with a PFE half-space having a soft reinforced layer at the interface. The free surface of the PFE layer is loaded with viscous liquid. The viscosity of the loaded liquid introduces losses and results in the damping of the wave. For the formulation of the problem with interface energy, the Gurtin–Murdoch approach is used. Using suitable conditions, a dispersion relation for propagating waves is derived in complex form. On separating the dispersion relation in real and non-real parts, the expressions relating the phase and damp velocities with wave number are derived. The obtained theoretical results are portrayed for the numerical data of PFE materials and distinct data of the interface layer. The obtained results are validated with pre-existing literature.

Anti-plane surface waves of an elastic half-space coated with a metacomposite layer

Anti-plane surface waves of an elastic half-space coated with a metacomposite layer

Tunable anti-plane wave bandgaps in 2D periodic hard-magnetic soft composites

In this paper, we present a theoretical model for the analysis of large magneto-deformation and the anti-plane shear wave bandgaps in 2D periodic two-phase hard magnetic soft composite structures subjected to magnetic stimuli. The constitutive behavior of the phases in the hard-magnetic soft composite is described using the incompressible Gent model. To solve the incremental anti-plane wave equations, the finite element method and the Floquet–Bloch theorem for periodic media are utilized. Using the developed framework, we numerically study the dependency of the bandgap width and their location on the direction and magnitude of the applied magnetic flux density vector, material parameter contrasts, and geometry and volume fraction of the inclusion phase. The numerical results reveal that significant tunability of the bandgap is achieved when the applied magnetic flux density direction is along the residual magnetic flux density direction. Also, it is seen that the geometry of the inclusion has significant effect on the bandgap width. The crucial inferences from the present investigation can find their potential use in the design and manufacturing of futuristic smart soft wave devices with tunable band structures.

Strain-gradient solution to elastodynamic scattering from a cylindrical inhomogeneity

In this paper, Mindlin’s first strain-gradient theory is adopted for the first time to come up with a solution to the problem of the elastic scattering of an anti-plane time-harmonic wave incident upon an isolated circular cylindrical inhomogeneity embedded in an infinite body. The effect of micro-inertia is also captured in this analysis, which alongside the strain-gradient effect enhances the adopted theory to account for different aspects of the size effect on the elastodynamic fields developed in the medium. The general case of an elastic inhomogeneity and two special cases of a rigid immovable obstacle and a cavity are addressed and the analytical expressions for the corresponding displacement fields are obtained. These solutions reveal the existence of some kind of decaying standing waves in the medium, which cannot be predicted by the classical elastodynamic theory. Moreover, the analytical expressions for the differential and total scattering cross-sections of the cylindrical inhomogeneity are determined. The results demonstrate that, if the cross-sectional radius of the inhomogeneity or the wave-length of the incident wave is comparable with the characteristic lengths of the medium, then the size effect will be significant and the discrepancies between the solutions obtained by the adopted theory and their counterparts in the classical elastodynamic theory will be pronounced.

Filtering property of periodic in-filled trench barrier for underground moving loads

The filtering properties of a 2-dimensional (2D) periodic in-filled trench barrier system subjected to anti-plane and in-plane moving loads are investigated separately by combining the spatial Fourier transform and periodic structure theory. By the aid of spatial Fourier transform, the governing equation of the periodic system is reduced in dimension. Based on the periodic structure theory, the closed-form solution of the dispersion equation for anti-plane wave is obtained; while for in-plane wave, the State-Space-Transfer-Matrix-Method (SSTMM) is introduced to solve the dispersion relation. Furthermore, the attenuation zones (AZs) varying with the transformed wave number, i.e., the AZs-kx graph, are obtained. Comparing the load-speed-line cluster of the moving excitation source with the obtained AZs-kx graph, the vibration isolation regions of the 2D periodic in-filled trench barrier in frequency-wave number space under moving loads can be found. To validate the present theoretical results, the performances of a 2D finite periodic in-filled trench barrier embedded in soil matrix and subjected to anti-plane and in-plane moving loads are numerically simulated, respectively. It is found that the AZs-kx graph plays a crucial role in revealing the isolation mechanism of the periodic in-filled trench barrier under moving loads. Theoretically speaking, it can be used to derive the vibration isolation regions for arbitrary excitation time histories of the moving-load inputs. The methodology proposed in the present paper can be used to design the continuous-type periodic wave barrier for isolating ambient vibration induced by moving loads.

Dynamic asymptotic homogenization for wave propagation in magneto-electro-elastic laminated composite periodic structure

In the present work, anti-plane wave propagation with oblique incidence in a magneto-electro-elastic multi-laminated composite periodic structure with perfect contact is studied. A general asymptotic homogenization method is applied to analyze dispersion phenomena and dynamic process. On assuming the solution having a single-frequency dependency, the higher-order terms for the displacement, electric potential and magnetic potential in asymptotic expansions are studied. Higher order local problems up to a finite order and a generalization of the local problem of infinite order are derived through rigorous calculations satisfying the necessary and sufficient condition for the existence of 1-periodic solutions. The effective equation of motion also known as “Good” Boussinesq equation has been achieved directly which is one of the highlight of the present work. Moreover, closed-form expression of dispersion equation and closed-form solutions of first and second local problems have been obtained. The analytical results have been validated for particular elastic case. Graphical illustrations have been done for tri-laminated magneto-electro-elastic composite periodic structure and the effect of size of unit cell, angle of incidence of wave, volume fraction of the composite and magneto-electric properties on the dispersion curve and also slowness curves have been examined. The numerical results have been validated with previous work and compared with results obtained by other approximation method. The present dynamic homogenization method has been proven to provide a more dispersive system and better approximation.

Anti-Plane Wave Scattering by a Circular Cavity in Complex Stratified Sites Based on the Hamilton System-Based Derivation of a Novel SBFEM

ABSTRACT For mathematical convenience, in scattering research, foundation soil is generally oversimplified to be uniform or horizontally stratified site. This study aims to evaluate anti-plane scattering by a circular cavity in complex stratified sites. To simulate the complex stratified sites, a novel scaled boundary finite element method (SBFEM) for scalar waves is first derived in the Hamilton system. The scattering analysis is performed utilizing the substructure replacement method. Numerical examples verify the accuracy of the method. Finally, the effects of the inclined and discontinuous stratified sites on the wave scattering of a circular cavity are investigated in frequency and time domains.

An analytical approach to model Structure–Soil–Structure Interaction (SSSI) of arbitrarily distributed buildings under SH waves

In this work, a simplified analytical framework based on the multiple scattering theory is proposed to model the structure–soil–structure interaction of buildings excited by antiplane shear waves. To this purpose, each building is modelled as a single degree-of-freedom oscillator, whereas the soil as a viscoelastic layer laying on an elastic half-space. By neglecting the soil-foundation kinematic interaction and considering only its inertial counterpart, the coupled response of buildings is modelled using a multiple scattering approach, where the buildings scattered wavefields are described via Green’s functions.The developed analytical framework is exploited to discuss the dynamic response of a single building, evaluating the variation of its amplitude with respect to the characteristic site frequencies. The dynamics of two buildings are then studied by modelling their coupled response. In particular, the interaction between them is investigated as a function of the buildings spacing, mass, and relative stiffness. Finally, the analysis is extended to the coupled response of a cluster of five buildings. Through the discussed examples, it is demonstrated how the proposed methodology can serve as a computationally inexpensive tool for predicting the interaction among vibrating structures under shear antiplane waves propagating at different frequencies.

Wave Propagation through a Piezoelectric Semiconductor Infinite Space

The propagation characteristics of anti-plane waves, i.e., the quasi-shear horizontal (QSH) waves and the carrier (CA) waves through a piezoelectric semiconductor (Piezo-SEMI) infinite space are discussed in this paper. First, the elastic, piezoelectric, dielectric, carrier mobility and diffusion constants in the considered propagation coordinate system are obtained by Bonde transformation from those in the crystal axis coordinate system. For anti-plane case, the secular equation with the mechanical, electric and the carrier concentration fields of the anti-plane waves through a Piezo-SEMI infinite space is derived. We obtain all possible wave modes, i.e., QSH waves and CA waves, that propagate in the Piezo-SEMI material by solving the secular equation. Then, the wave speeds and attenuation of QSH waves and CA waves are obtained. The steady-state carrier concentration and biasing electric field have significant effects on dispersion and attenuation of the QSH waves and the CA waves, and the sensitive areas to the wave velocity and attenuation are obtained.

Deep learning-based design of ternary metamaterials for isolating full-mode waves

A deep learning-based methodology is presented to design the topologies and periodic constants of ternary metamaterials under different site conditions, considering both in-plane and anti-plane waves (full-mode waves) coming together. A variational autoencoder (VAE) and a series–parallel neural network (SPNN) are constructed, based on which the design model is established and the metamaterial customization is realized. One thousand designs are performed, where the determination coefficient reaches 0.982, the root mean square error is only 0.785, and the mean design error is 3.1%, evaluating the accuracy and stability of the deep learning method. To illustrate the generality of the presented method, metamaterials under different site conditions are designed. For the same target, multiple sets of metamaterials are given, which reflects the non-uniqueness of design problems. Two sets of metamaterials respectively for two practical examples are customized and discussed with 3D finite element models, verifying their ability of isolating vibrations in all directions.

Micro-structural effects in phononic dielectric structures

Based on the higher-grade continuum theory, propagation of longitudinal as well as transverse anti-plane elastic waves in normal direction to nanoscale periodic laminates of piezoelectric dielectrics is studied in this paper. The strain gradients, micro-inertia and direct flexoelectricity phenomena are incorporated into the phenomenological description. The problem is analysed as one-dimensional and the governing equations together with possible boundary conditions are derived from the Hamilton variation principle. It is shown that the transverse waves are not affected by the electric polarization in contrast to the longitudinal waves. The developed formulation is applied to 1D Bloch waves obeying perfect boundary conditions periodical in bi-material laminates and the transfer matrix method is used for derivation of dispersion equation. An analytical method is developed for solution of the dispersion equation. The influence of the micro-stiffness and micro-inertial length scale parameters as well as flexoelectric coefficients on the dispersion curves and the frequency gaps is investigated in parametric study.

A mathematical analysis of anti-plane surface wave in a magneto-electro-elastic layered structure with non-perfect and locally perturbed interface

In the present work, an analytical method has been adapted in order to study dispersive behaviour of anti-plane surface wave, in particular, Love wave in a homogeneous fully-coupled magneto-electro-elastic layer/half-space configuration. The interface is considered mechanically, electrically and magnetically non-perfect as well as slightly non-parallel where a geometrically perturbed region of finite span is considered to be small compared to the wavelength of observed phases. Closed-form expressions of dispersion equations for rectangular-shaped irregularity and non-perfect interface existing in magneto-electro-elastic layered structure have been derived analytically for four different types of electromagnetic boundary conditions. Eringen’s linear perturbation method and Fourier transformation have been used and the obtained results have been validated with the solutions for simpler layered structures. Two laminated composites BaTiO3/CoFe2O4 of different volume fractions have been considered in order to plot the dispersion curves in distinct cases. The effect of irregularity and mechanical, electrical and magnetic non-perfect bonding on phase velocity and magneto-electromechanical coupling has been studied. The present study holds potential applications in multidisciplinary fields, such as designing acoustic devices, magnetic field sensors and actuators, in particular, Love-wave SAW magnetic-field sensors as well as in seismology when seismic signals propagate through Earth medium consisting of magnetic materials.

Transference of SH waves in a piezoelectric fiber-reinforced composite layered structure employing perfectly matched layer and infinite element techniques coupled with finite elements

Most commercially used piezoelectric materials suffer from certain drawbacks like low piezoelectric output, brittle nature, etc. Thus, they cannot be utilized to their fullest extent in dynamic applications involving actuators, transducers, Surface Acoustic Wave (SAW) devices, and Love wave sensors, among others. The present work focuses on the transference attributes of anti-plane SH wave in a layered structure composed of a piezoelectric fiber-reinforced composite (PFRC) layer lying over a piezoelectric half-space. The PFRC composed of PZT-5A-epoxy is micro-mechanically modeled using Strength of Materials and Rule of Mixtures, and some of its electro-mechanical advantages over monolithic PZT-5A are discussed. The dispersion relation is obtained by analytical means, as well as two numerical techniques, viz., Semi-Analytical Finite Element - Perfectly Matched Layer (SAFE-PML) technique and Semi Analytical Infinite Element (SAIFE) technique with (1/r) type decay. The Rayleigh–Ritz method and Gauss 3-point quadrature formula are employed to derive the dispersion relation numerically. The effects of the existing parameters, viz. fiber volume fraction, PML length, PML function, etc., on the dispersion curves are graphically demonstrated and discussed. The present work is validated by matching the obtained results with the ones found in the extant literature.

Numerical simulation of anti-plane wave propagation in heterogeneous media

This paper formulates a semi-analytical boundary meshless method, the singular boundary method (SBM), to simulate the anti-plane wave motion in the heterogeneous media. Two different continuous inhomogeneity variations are studied, and the corresponding fundamental solutions are derived by means of a simple variable transformation. The derivation reveals that the fundamental solutions are determined by the wavenumbers of the resulted governing equations after the variable transformation. The source singularities of these fundamental solutions are overcome by the novel origin intensity factors (OIFs). The fundamental solutions and OIFs make the SBM applicable for the simulation of the anti-plane wave scattering and diffraction in nonhomogeneous media. Numerical experiments, including the finite and semi-infinite cases with different variations and boundary conditions, are presented to illustrate the convergence, accuracy and effectiveness of the proposed SBM.

Anti-Plane Wave Propagation in the Functionally Graded Hybrid Structure under an External Impulsive Force: A Green’s Function Approach

To propagate the anti-plane wave, a hybrid layered-half-space schematic of fiber-reinforced and orthotropic materials is taken into account. Due to its importance in engineering applications, anti-plane wave propagation through functionally graded orthotropic materials has attracted a lot of attention. The functionally graded orthotropic materials in the half-space and the reinforced materials under high initial stress are utilized in the superficial layer of finite depth. To examine the effect of heterogeneity on wave phase velocity, the elastic characteristics of orthotropic materials are defined in terms of hyperbolic functions. Moreover, a rectangular plate is installed at the interface of the materials to expose the effect of irregularity in the materials. An external force (impulsive force) is applied to the wave at the point source of the disturbance to understand the influence of such a force on the phase velocity of the wave. The nonhomogeneous equations of motion are deduced by Fourier transformation and are solved analytically by Green’s function technique along with possible applications of Dirac delta function. The dispersion relation of the anti-plane wave propagation is obtained analytically. This study presents a graphic explanation of the proposed schematic and discusses the stability of the model.