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Film/substrate structures are often seen in nature and engineering and their characteristic mechanical behaviors facilitating widespread application prospects. Surface patterns of film/substrate structures significantly affect the propagation of elastic waves. This paper investigates elastic wave propagation in the thin film/substrate periodic structure with sinusoidal surface patterns through the uniaxial tension. Based on the Bloch wave theory and numerical simulation, diverse bandgaps are triggered and the overall frequency range of the bandgaps shifts upward with the increase of the tensile strain, which demonstrates that tensioning the bilayer is a reliable and robust way for manipulating bandgaps. Band properties and bandgap evolution by various geometric and material parameters are discussed systematically, including various surface sinusoidal amplitudes, thickness of the substrate and various density or modulus ratio. The underlying mechanism of the exciting results in reversal design and regulation of structural geometry is revealed and is explained by using the spring-mass model. This work can provide new thought and insight for controlling wave propagation and promoting potential applications of the film/substrate systems and devices. [ABSTRACT FROM AUTHOR]

Bandgaps are intrinsic physical properties of periodic structures, for example, mechanical metamaterials or phononic crystals, which indicate attenuation of wave propagation within a specific frequency range. Bandgap calculation of mechanical metamaterials is realized via discretizing a representative voluame element (RVE) and imposing complex-valued Bloch wave boundary conditions. It is highly desirable to implement Bloch wave analysis in real-valued commercial finite element software. We present and detail an ABAQUS and Python implementation of Bloch wave analysis. Two-dimensional (2D) soft as well three-dimensional (3D) hard mechanical metamaterials are chosen as numerical examples. Material and geometric nonlinearities are tackled properly and the effects of pre-strains are explored. Our numerical results are validated either by similar results reported in the literature or by our experimental measurement. The presented numerical strategy would aid the design, analysis and utilization of mechanical metamaterials for dynamic applications. The codes and data of the paper could be downloaded from https://github.com/XJTU-Zhou-group/Calculating-band-gaps-of-nonlinear-mechanical-metamaterials. [ABSTRACT FROM AUTHOR]

This article presents a multifield asymptotic homogenization scheme for the analysis of Bloch wave propagation in non-standard thermoelastic periodic materials, leveraging on the Green-Lindsay theory that accounts for two relaxation times. The procedure involves several steps. Firstly, an asymptotic expansion of the micro-fields is performed, considering the characteristic size of the microstructure. By utilizing the derived microscale field equations and asymptotic expansions, a series of recursive differential problems are solved within the repetitive unit cell Q. These problems are then expressed in terms of perturbation functions, which incorporate the material's geometric, physical, and mechanical properties, as well as the microstructural heterogeneities. The down-scaling relation, which connects the microscopic and macroscopic fields along with their gradients through the perturbation functions, is then established in a consistent manner. Subsequently, the average field equations of infinite order are obtained by substituting the down-scaling relation into the microscale field equations. To solve these average field equations, an asymptotic expansion of the macroscopic fields is performed based on the microstructural size, resulting in a sequence of macroscopic recursive problems. To illustrate the methodology, a bi-phase layered material is introduced as an example. The dispersion curves obtained from the non-local homogenization scheme are compared with those generated from the Floquet-Bloch theory. This analysis helps validate the effectiveness and accuracy of the proposed approach in predicting the wave propagation behavior in the considered non-standard thermoelastic periodic materials. [Display omitted] • Multifield asymptotic homogenization scheme in non-standard thermoelasticity. • Bloch wave propagation in periodic Cauchy materials is investigated. • The Green-Linsdsay model accounting for two relaxation times is leveraged. • Higher order approximations of complex frequency band structures are obtained. • Illustrative examples, validating the accuracy of the approach, are performed. [ABSTRACT FROM AUTHOR]

The dynamics of rotating systems plays an important role in the efficiency and reliability of such machines. For this reason, understanding their dynamics is essential to predict and mitigate the resultant vibrations that can occur in such machines. Considering that vibration is essentially the propagation of waves in a continuous medium, methods of wave analysis can also be employed to investigate the dynamics of rotating systems, especially when the system presents some sort of periodicity. The difficulty lies in the fact that rotating systems are non-reciprocal systems due to the gyroscopic effects, which create forward and backward whirl motions with associated mode waves. The conventional wave methods usually do not tackle this problem. This work presents a methodology for analyzing rotating systems with wave methods based on the combination of conventional modal analysis, Bloch wave propagation theory, and directional frequency response functions. In this case, it is possible to separate the forward and backward modes, thus allowing the analysis of forward and backward wavenumbers separately. Consequently, one can draw separate dispersion diagrams for the wave modes associated with the forward and backward motions of the rotor. The numerical results show the feasibility of the method, and the methodology is applied experimentally to a rotating system in the laboratory. [Display omitted] • Wave analysis of a periodic rotor considering gyroscopic effects. • Forward and backward whirl modes belong to different wave groups. • Band gaps can differ from non-propagating and strongly attenuated propagation. • Wave Campbell diagram separates the modes and show the respective wavenumbers. [ABSTRACT FROM AUTHOR]

Many natural and human-made material systems (e.g., bone, shale, and cement-based composites) exhibit heterogeneous microstructures. Lattice models have reemerged to simulate such material systems because of their inherent simplicity while offering tremendous capabilities. However, current lattice models suffer from several deficiencies; for example, some lattice models cannot span the necessary range of Poisson's ratio, some others are not isotropic (e.g., the standard square lattice), and others are not practical for complex domains (e.g., equilateral triangular lattice and hexagonal lattice). Thus, there is a need for a simple lattice that can handle Poisson's ratio without any limitations, capture all the possible deformation modes of an isotropic elastic material, have minimal degrees-of-freedom, and provide positive definite stiffness and mass matrices. In this paper, we develop such a lattice model. Our approach hinges on equating the Lagrangians of the continuous (continuum) and discrete (lattice) systems, defining the strains consistently in terms of displacements in the lattice, adding a local interaction term to span the entire invariant space, and using an energy preserving time-stepping scheme. Using a Bloch wave analysis, we show that the lattice is, in fact, isotropic. Also, the lattice model does not suffer from volumetric locking, which is not the case with low-order finite elements. We verify the accuracy of the model using analytical solutions on benchmark problems. Finally, we demonstrate the application of the model on a practical example by performing propagation analysis in cement paste microstructure acquired from scanning electron microscopy (SEM). [ABSTRACT FROM AUTHOR]

The photonic spin Hall effect (PSHE) can be realized in a photonic crystal (PC) slab, that is, the unidirectional Bloch surface wave can propagate along the surface of the PC slab under the excitation of elliptical polarized magnetic dipole. It is further proved that PSHE is caused by the interference of the component surface waves excited by the different components of the incident light, which is the so called component wave interference (CWI) theory. In addition, we also find that the spin of the surface wave oscillates periodically in space, and the oscillation period is a unit cell. In a unit cell, the average spin keeps the spin orbit locked. The results show that the spin separation can also be modulated by the position and the polarization state of the magnetic dipole. [ABSTRACT FROM AUTHOR]

Magneto-electro-elastic (MEE) heterogeneous materials with periodic microstructure are investigated, with special focus on the particular case of a layered MEE material. By exploiting the transfer matrix method in combination with the Floquet–Bloch boundary conditions, the frequency dispersion spectrum can be derived by solving an eigenproblem, involving a 12 × 12 transfer matrix with palindromic characteristic polynomial. In particular, by assuming the cubic symmetry of layers, the problems involving longitudinal and transverse elastic waves are uncoupled. This circumstance has the advantage of simplifying the treatment and the size of the subproblems to be solved, corresponding to a 4 × 4 transfer matrix for the longitudinal case, and a 8 × 8 transfer matrix of the transverse problem, both characterized by palindromic characteristic polynomials. Afterward, a dynamic multi-field continualization approach is proposed to investigate wave propagation in MEE layered periodic material by matching the Z -transform of the vector collecting the nodal fields to the two-sided Laplace transform of the same vector at the macroscale. The continualization technique allows identifying multi-field integral-type non-local continua as well as multi-field generalized gradient-type (higher-order) non-local continua. Finally, to verify the accuracy of the proposed approach, the dispersion curves derived from the continualization technique are compared with the curves provided by the Floquet–Bloch theory. • Magneto-Electro-Elastic periodic microstructured materials are investigated. • Transfer matrix method is applied to obtain complex frequency dispersion spectra. • A dynamic multi-field continualization scheme of layered MEE is conceived. • Multi-field generalized gradient-type (higher-order) non-local continua are identified. [ABSTRACT FROM AUTHOR]

Wrinkling is a common phenomenon in natural and engineering film structures. The wrinkles influence the geometry and dynamic response of these structures. In this work, we investigate the wrinkling of a stretched thin film containing engineered microstructures and its derived functionality on controlling the propagation of bending waves. The underlying mechanism is revealed and the effect of wrinkles on the bandgap of bending waves is systematically evaluated via numerical simulations based on the Bloch wave theory. We show that wrinkles with a customized wavelength can be triggered in the microstructured film due to the mismatched deformation in the film. The bandgap of the wrinkled film can be finely tuned via applied stretching, resulting in the controllable propagation of bending waves in thin films. Our work provides fundamental insights into wave propagation in wrinkled films and potential applications for dynamic control of the wave propagation in engineering film structures. [ABSTRACT FROM AUTHOR]

This work examines elastic wave propagation phenomena in open-cell foams with the use of the Bloch wave method and finite element analysis. Random foam topologies are generated with the Surface Evolver and subsequently meshed with Timoshenko beam elements, creating open-cell foam models. Convergence studies on band diagrams of different domain sizes indicate that a representative volume element (RVE) consists of at least 8³ cells. Wave directionality and energy flow features are investigated by extracting phase and group velocity plots. Explicit dynamic simulations are performed on finite size domains of the considered foam structure to validate the RVE results. The effect of topological disorder is studied in detail, and excellent agreement is found between the wave behavior of the random foam and that of both the regular and perturbed Kelvin foams in the low-frequency regime. In higher modes and frequencies, however, as the wavelengths become smaller, disorder has a significant effect and the deviation between regular and random foam increases significantly. [ABSTRACT FROM AUTHOR]

Periodic lattices offer enhanced mechanical and dynamic properties per unit mass, and the ability to engineer the material response by optimizing the unit cell. Characterizing the effective properties of these lattice materials through experiments can be a time consuming and costly process, so analytical and numerical methods are crucial. Specifically, the Bloch-wave homogenization approach allows one to characterize the effective static properties of the lattice while simultaneously analyzing wave propagation properties such as band gaps, propagating modes, and wave directionality. While this analysis has been used for some time, a thorough study of this approach on three-dimensional (3D) lattice materials with different symmetries and geometries is presented here. Bloch-wave homogenization is applied to extract the effective stiffness tensor of 3D periodic lattices and confirmed with elastostatic homogenization, both within a finite element framework. Multiple periodic lattices with cubic, transversely isotropic, and tetragonal symmetry, including an auxetic geometry, over a wide range of relative densities are analyzed. Further, this approach is used to analyze 3D periodic composite structures, and a way to tailor their overall anisotropy is demonstrated. This work can serve as the basis for nondestructive evaluation of metamaterials properties using ultrasonic velocity measurements. [ABSTRACT FROM AUTHOR]

Abstract The stop and pass bands (i.e. the band gaps) characteristics are determined for gyroscopic systems by developing an approach which is compatible with such class of systems which is based on the concept of Bloch wave propagation in periodic structures. In this approach, the dispersion curves of the periodic gyroscopic systems are determined for different rotational speeds. The obtained characteristics are compared with non-rotating systems in an attempt to quantify the effect of the gyroscopic forces on the “ band gap ” characteristics. The developed approach is illustrated by a new class of drill strings with passive periodic inserts. These inserts are utilized to filter out the vibration transmission along the drill string. Such mechanical filtering capabilities allow the vibrations to propagate along the periodic drill string only within specific frequency bands called the ‘ pass bands ’ and completely block it within other frequency bands called the ‘ stop bands ’. The inserts introduce impedance mismatch zones along the vibration transmission path to impede the propagation of vibration along the drill string. The design and the location of the inserts are selected to confine the dominant modes of vibration of the drill string within the stop bands generated by the periodic arrangement of the inserts in order to completely block the propagation of the vibrations. A finite element model (FEM) that simulates the operation of this new class of drill strings is developed to describe the complex nature of the vibration encountered during drilling operations. The FEM is used to extract the dispersion characteristics of the gyroscopic unit cell of the drill string in order to determine its stop and pass band characteristics. Experimental prototype of the passive periodic drill string is built and tested to demonstrate the feasibility and effectiveness of the concept of periodic drill string in mitigating undesirable vibrations. The experimental results are used to validate the developed theoretical model in order to develop a scalable design tool that can be used to predict the dynamical behavior of this new class of drill strings. [ABSTRACT FROM AUTHOR]

Propagation of high-frequency wave in periodic media is a challenging problem due to the existence of multiscale characterized by short wavelength, small lattice constant and large physical domain size. Conventional computational methods lead to extremely expensive costs, especially in high dimensions. In this paper, based on Bloch decomposition and asymptotic analysis in the phase space, we derive the frozen Gaussian approximation for high-frequency wave propagation in periodic media and establish its converge to the true solution. The formulation leads to efficient numerical algorithms, which are presented in a companion paper [SIAM J. Sci. Comput.38 (2016), A2440–A2463]. [ABSTRACT FROM AUTHOR]

The periodic cellular topology characterizing the microscale structure of a heterogeneous material may allow the finest functional customization of its acoustic dispersion properties. The paper addresses the free propagation of elastic waves in micro-structured cellular materials. Focus is on the alternative formulations suited to describe the wave propagation in the material, according to the classic canons of solid or structural mechanics. Adopting the centrosymmetric tetrachiral microstructure as prototypical periodic cell, the frequency dispersion spectrum resulting from a synthetic lagrangian beam-lattice formulation is compared with its counterpart derived from different continuous models (high-fidelity first-order heterogeneous and equivalent homogenized micropolar continuum). Asymptotic perturbation-based approximations and numerical spectral solutions are cross-validated. Adopting the low-frequency band gaps of the material band structures as functional targets, parametric analyses are carried out to highlight the descriptive limits of the synthetic models and to explore the enlarged parameter space described by high-fidelity models. The final tuning of the mechanical properties of the cellular microstructure is employed to successfully verify the wave filtering functionality of the tetrachiral material. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Meta-structures with dynamic vibrational resonators (DVRs) are programmed to control the propagation of waves and attenuate vibrations over a broadband frequency spectrum. Attributes of DVRs, such as their resonant frequency and mass, determine the location and width of the bandgap, respectively. As a result, to adaptively program bandgaps, one has to modify or tune the eigenvalues of individual DVRs, and a popular approach is to vary the stiffness of each resonator. However, the tunable range of bandgaps is often restricted to maximum change in DVRs' stiffness. This work presents a novel approach to adaptively program bandgaps of a 1D flexural meta-structure over a broad frequency bandwidth. DVRs with two stable configurations are attached to a beam in developing the meta-structure. A numerical model is developed to illustrate the scope of the novel approach. An experimental investigation then validates the simulated results and shows the extent of the vibration absorption capabilities of the meta-structure. A reinforced learning approach is used to adaptively tune the bandgap over 220 Hz to 840 Hz. • Various configurations of meta-structures are introduced with bistable resonators. • Multiple bandgap formation is explained by introducing patterns of n-bit unitcells. • Bandgap locations for various unitcell patterns is simulated by Bloch wave analysis. • Validation of Finite element formulation is conducted by experiments. • RL is implemented to switch between different configurations of meta-structure. [ABSTRACT FROM AUTHOR]

Propagation of anti-plane waves through a discrete square lattice and through a continuous fibrous medium is studied. In the long-wave limit, for periodically heterogeneous structures the solution can be periodic or anti-periodic across the unit cell. It is shown that combining periodicity and anti-periodicity conditions in different directions of the translational symmetry allows one to detect different types of modes that do not arise in the purely periodic case. Such modes may be interpreted as counterparts of non-classical waves appearing in phenomenological theories. Dispersion diagrams of the discrete square lattice are evaluated in a closed analytical from. Dispersion properties of the fibrous medium are determined using Floquet-Bloch theory and Fourier series approximations. Influence of a viscous damping is taken into account. [ABSTRACT FROM AUTHOR]

The paper presents an efficient and accurate algorithm to analyze the fluid/solid and solid/fluid phononic crystals (PCs) with fluid structure interaction (FSI). In the proposed mass-redistributed finite element method (MR-FEM), the dispersion error in the longitudinal wave of fluid domain is minimized by re-distributing the mass elements in the mass matrix to “tune” the balance, while the lumped mass matrix is employed to analyze the elastic wave within the solid. In addition, different Bloch boundary conditions including the square and triangular lattice using MR-FEM are also formulated in this work. The accuracy of the proposed MR-FEM is verified by using the commercial software COMSOL, and the computational efficiency of MR-FEM is significantly greater than the conventional finite element methods (FEM) based on six different numerical examples. The MR-FEM with excellent features has a great potential to extend 3D multi-physics domain of PCs. [ABSTRACT FROM AUTHOR]

The Bloch mode synthesis (BMS) model-reduction method adapts component mode synthesis techniques to unit-cell problems in order to obtain a reduced-order model that quickly produces band-structure frequencies for any wave vector, or vice versa. Fundamental to BMS is a partitioning of the real-space model into interior and boundary components, and subsequent reduction of the interior via truncated normal mode expansion. In this paper, two enhancements are presented for the BMS method that reduce both computation time and error in band-structure calculations. The first enhancement improves the accuracy of the interior reduction by approximating the participation of the residual modes rather than simply truncating them. The original formulation of BMS includes a modal reduction of the boundary that must be recomputed for every wave vector. This limits computational benefits and prevents the reduced-order model from being useful for the inverse band-structure problem (i.e., the k ( ω ) calculation). The second enhancement is a local boundary reduction that is independent of wave vector and thus does not suffer from the aforementioned limitations. [ABSTRACT FROM AUTHOR]

In-plane wave propagation in hexagonal and re-entrant lattices is a widely investigated subject in the literature. Such systems can exhibit many different but limited band structure properties that depend on the topology and geometry of structural members and the lattice itself. This manuscript proposes a novel class of hexagonal and re-entrant lattices with unit cells containing combined straight and curved beams with enhanced band-gap properties. Timoshenko beams are suggested to represent beam members of the lattice and corresponding governing equations are derived. Bloch theorem is applied to study in-plane wave propagation and get the unique dispersion properties of the modified lattices. The influence of a new geometric parameter, the curvature angle of the constituent curved beams is explored for the dispersion characteristics and wave directionality through iso-frequency contours of dispersion surfaces. Enhanced band-gap properties in the context of the generation of new band-gaps and widening of the band-gaps for the cellular lattices with curved beams are promising from the viewpoint of the future design of phononic crystals and metamaterials in their mechanical setup. [ABSTRACT FROM AUTHOR]

This paper investigates coupled electro-elastic shear waves propagating along a piezoelectric finite width waveguide consisting of layers separated by metallized interfaces and arranged in a periodic way along the guide. The modified matrix method is applied to obtain the dispersion equation for a waveguide with straight walls. Bragg resonances and the presence of trapped modes and slow waves are revealed and analyzed for a periodic structure consisting of unit cells made up firstly from two different piezoelectric materials, and secondly from two identical piezoelectric materials. An analytical expression for the transmission coefficient for the waveguide with a defect layer is found that can be used to accurately detect and control the position of the passband within a stopband. This can be instrumental for constructing a tuneable waveguide made of layers of identical piezoelectric crystals separated by metallized interfaces. [ABSTRACT FROM AUTHOR]

We describe in this work a composite metamaterial with a hierarchical topology made by tessellating perforations that exhibit an auxetic (negative Poisson’s ratio) behaviour. We perform an analysis of the hierarchical structure by evaluating the fractal order of the topologies associated to the perforated composites.The periodic hierarchical lattice configuration shows negative Poisson’s ratio characteristics at higher levels of hierarchy, even when the baseline configuration has a topology not exhibiting an auxetic behaviour. We investigate the wave propagation characteristics of these particular hierarchical lattices by using a Bloch wave approach applied to detailed Finite Element geometries of the unit cell configurations. We show that the level of hierarchy creates new band gaps with large relative widths, and it also shifts the same bandgaps towards lower frequencies. We correlate the mechanical properties, fractal order and the dispersion characteristics of the multiscale auxetic perforated metamaterial with the parameters defining the geometry of the lattice and the hierarchy levels, and discuss the results in a nondimensional form to provide a performance map of the mechanical and dynamic properties. [ABSTRACT FROM AUTHOR]

The problem of shallow-water wave propagation over an infinite array of periodic trapezoidal bars and triangular bars is studied. By employing the linear shallow-water wave theory and the well-known Bloch theorem, the eigenvalue problem in terms of the wave number for given wave frequency, water depth, and geometric properties of the bars is formed. A closed-form solution of the eigenvalue problem is derived which breaks the restriction of piecewise constant water depth required in almost all of the previous analytic solutions. The solution is verified against the existing solution for the special case of rectangular bars. Based on the present solution, gap maps in various cases are plotted which exactly show the distribution of band gaps for wave propagation over both trapezoidal bars and triangular bars, and the influence of the given wave frequency and the geometric parameters of bars such as height and width on the occurrence of band gaps is analyzed. By using the gap maps presented in the paper, the condition under which the waves can be completely blocked by an infinite array of trapezoidal bars and triangular bars can be easily and exactly determined. [ABSTRACT FROM AUTHOR]

Magneto-electro-elastic (MEE) materials are attracting an increasing amount of scientific interest in physics, for their valuable potential in the functional management of mechanical-to-magnetoelectric energy conversions. The present paper focuses on heterogeneous MEE media – artfully realized by virtue of the periodic repetition of a multi-phase elementary cell – which represent promising solutions to passively control the propagation of multifield Bloch waves. Specifically, the linear continuum model governing the coupled free dynamics of a cellular non-dissipative MEE material is formulated for the unbounded three-dimensional domain. Then, the governing equations are specialized for a periodic layered waveguide with perfect inter-layer interfaces. The Floquet–Bloch decomposition is employed to analytically determine the uni-modular transfer matrix of the heterogeneous multi-layered cell. The related eigenproblem, governed by a palindromic characteristic polynomial in the Floquet multipliers, is solved in closed form to analyze the dispersion properties in the real-valued frequency domain and complex-valued wavenumber domain. Alternatively, a perturbation scheme is outlined to asymptotically approximate the exact eigensolutions. The dispersion spectra, composed of propagation and attenuation branches, are parametrically investigated for a two-layered periodic waveguide. The corresponding band structures are assessable a priori by leveraging the formal analogy with the stability analysis for non-autonomous dynamical systems. As main findings, slow-propagating quasi-shear-elastic waves and fast-propagating quasi-magneto-electric waves are found to dominate the low-frequency and high-frequency ranges, respectively, although some crossing points between the respective spectral branches can be detected. Quantitatively different but qualitatively similar behaviors of propagation and/or attenuation are observed for the quasi-shear-elastic waves and quasi-magneto-electric waves. Consequently, pass–pass, pass–stop and stop–stop bands of frequencies can be recognized and tuned by varying the key physical parameters. This rich spectral scenario opens the way for the functional customization of MEE-based metafilters or metapropagators, purposely designed to govern the transfer, localization, conversion, harvesting of energy across large frequency ranges. • Free wave propagation through a periodic non-dissipative MEE material is studied • The dispersion properties of periodic layered waveguide are analytically derived • Floquet multipliers are derived by solving the palindromic characteristic polynomial • Perturbation scheme is outlined to asymptotically approximate the eigensolutions • Pass–pass, pass–stop and stop–stop bands of frequencies can be recognized and tuned [Display omitted] [ABSTRACT FROM AUTHOR]

• A theoretical model is proposed for Bloch waves in a soft lattice. • The coupled modes are converted into easy modes by finite stretching. • Mode evolution is governed by effective shear modulus and length of the component. • The broadest directional bandgap is observed for equi-biaxial stretch. • Theoretically predicted flat bands do not exist in real soft lattices. In this study, the in-plane Bloch wave propagation in a finitely stretched square lattice is investigated numerically and theoretically. Precisely, the elastic band diagram is calculated for an infinite periodic structure with a cruciform hyperelastic unit cell under uniaxial or biaxial tension. Meanwhile, an elastodynamic tight-binding model is proposed to investigate the formation and evolution of the band structure. In said lattice, the finite stretch was found to not only tune the symmetry of the band structure but also purify the elastic waves into easy modes. The uniaxial stretch exhibits the opposite impacts on the evolution of two easy modes, while the effect of the biaxial stretch is equated with the superposition of the uniaxial stretches in the two tessellation directions. The mentioned effects on the band structure could be attributed to the competition between the effective shear moduli and lengths for different beam components. As a result, the finite stretch could tune the directional elastic wave bandgap of the soft lattice, and the broadest bandgap could be anticipated in an equi-biaxial stretch. This study may shed some light on the design and implementation of elastic wave control devices with weight efficiency and tunability. It also reveals that the theoretically predicted flat bands did not exist in the numerical calculations, a difference between the physical system and the corresponding simplified theoretical model. [ABSTRACT FROM AUTHOR]

Propagation of elastic waves in discontinuous media is studied in this paper by the scattering matrix method (SMM). An electromagnetic transmission line analogy is also used to set up the mathematical model. The SMM operates in the frequency domain and allows for all wave polarizations (P, SV and SH). Rock masses are examples of discontinuous media in which the discontinuities (fractures or joints) influence wave propagation. Both elastic and viscoelastic joints are considered and the latter are described by Kelvin-Voigt, Maxwell and Burgers models. Rock joints with Coulomb slip behavior are also analyzed, by applying the averaging principle of Caughy (J Appl Mech 27:640-643, ). The evaluation of the effects of periodic discontinuities in a homogeneous medium is presented by introducing the concept of Bloch waves. The dispersion curves of these waves are useful to explain the existence of frequency bands of strong attenuation, also in the case of lossless (perfectly elastic) structures. Simple expressions of transmission and reflection coefficients are obtained. Finally, the SMM results are compared with those computed via the distinct element method (DEM). The comparisons are performed on a medium with joints with Coulomb slip behavior and the agreement is satisfactory, although the SMM must be applied in conjunction with the equivalent linearization technique. Even if the DEM is much more general, the SMM in these simple cases is extremely faster and provides a higher physical insight. [ABSTRACT FROM AUTHOR]

• Band structure of elastically connected periodic beams system is determined. • The unit cells are organized like the diatomic and triatomic chain. • Frequency response functions are used to validate the obtained dispersion curves. • Effects of medium stiffness and point masses on the band structure are investigated. This paper proposes the methodology to carry out the analysis of Bloch wave propagation in an array of vertically aligned and elastically connected structural elements such as beams, strings, plates, or other slender structures. The suggested approach is based on the Galerkin approximation and Floquet-Bloch theorem used in defining the eigenvalue problem and obtaining the band structure of the periodic systems. Special attention is devoted to the case of elastically connected Rayleigh beams with attached concentrated masses and wave propagation in the direction normal to the beam's length. A validation study is performed by using the finite element model and the frequency response function to confirm the accuracy of the solution obtained via the Galerkin approximation. Two configurations of unit cells, having two and three elastically connected beams with different geometrical and material properties, are considered in the numerical study. The effects of various parameters are investigated to reveal their influence on the frequency band structure and emergence of the zero-frequency bandgap. The results of this study demonstrates the tunability properties of the proposed periodic systems due to changes in values of concentrated masses, stiffness of the coupling medium or boundary conditions on structural elements within the unit cell. [ABSTRACT FROM AUTHOR]

In this study, the periodic nested acoustic black hole phononic structure is presented, and different complex band structures and corresponding evanescent Bloch wave propagation are discussed. Vibration transmissions and modes indicated that the corresponding regions of the bandgaps were due to the coupling effect between the Fano-like resonance and Bragg scattering, which also presented the negative mass and stiffness characteristics. Moreover, strain tensors at interface and central sections of the unit cells were used to evaluate the structural instability and failure. Contrast models proved the superiority of the periodic nested acoustic black hole phononic structure for evanescent Bloch wave control and vibration experiments verified its effectiveness. The relevant conclusions can be used in the vibration reduction design of lightweight structure and enrich the relevant research experience on acoustic black hole. [ABSTRACT FROM AUTHOR]

• Complex band structures are calculated by considering partial open-pore interfaces. • The first avoided crossing vanishes with the increase of pore blockage coefficient. • The first avoided-crossing becomes wide with the increase of the porosity. Complex band structures of the longitudinal wave propagation normally in fluid-saturated porous periodic structure with partial open-pore interfaces are calculated. The results show that the first avoided crossing gradually vanishes with the increase of the pore blockage coefficients. [Display omitted] The propagation of waves in fluid-saturated porous periodic structures is significantly affected by the interface condition between adjacent layers. We consider in this paper the partial-open pore interface condition between adjacent layers in a one-dimensional fluid-saturated porous phononic crystal. A transfer matrix method is devised to obtain both the complex band structure and the poroelastic Bloch waves of the crystal. Spectral transmission through a finite structure is further computed by a stiffness matrix method. Attention is restricted to normal incidence of longitudinal waves. The influence of the pore blockage, a parameter of the partial-open pore interface condition, and of porosity and viscosity are investigated. The value of the pore blockage is found to influence significantly both the dispersion of poroelastic waves but also the partition of wave energy between solid skeleton and pore fluid. The effects of porosity and viscosity in the case of the partial-open pore interface condition are similar to what was previously obtained in the fully open pore case. [ABSTRACT FROM AUTHOR]

• The Dirichlet-to-Neumann map method is extended to compute the band structures of 2D viscoelastic phononic crystals where the shear modulus of the viscoelastic matrix material is dependent of the frequency. • The relaxation time and the final-initial modulus ratio determine the main influence domain of the viscosity and the amplitude of the shear modulus. • The complex band structures are computed for different viscosity parameters and the wave propagation and attenuation with the change of the viscosity are analyzed. • Based on the storage shear modulus, the band structures are studied and compared with that of the purely elastic case. • The effects of various parameters on the band structures of viscoelastic phononic crystals are discussed. The effects of viscosity on the band-baps of two-dimensional (2D) viscoelastic phononic crystals are studied by the Dirichlet-to-Neumann (DtN) map method. The viscoelastic phononic crystals are composed of square or triangular lattices of elastic circular solid cylinders in a viscoelastic solid matrix. Time-harmonic anti-plane transverse (SH) waves are considered and the standard linear solid model is used to describe the viscoelastic behavior of the matrix material, in which the shear modulus depends on the frequency. Based on the DtN map constructed by using the cylindrical wave expansions, a linear eigenvalue problem related to the Bloch wave vector is formulated, which involves relatively small matrices. By using real frequencies and complex-valued wave vectors, the complex band structures are computed for different viscosity parameters, and the wave propagation and attenuation characteristics in dependence of the viscosity are analyzed. Then based on the storage shear modulus, real band structures for the Pb/epoxy phononic crystals with viscosity in the square and triangular lattices are studied, and the effects of the different viscoelastic material parameters and other relevant influencing parameters on the band-gaps are discussed. The results show that the viscosity of the matrix material affects the location and the width of the band-gaps. This fact provides an alternative way to tune the band-gaps of phononic crystals by adjusting the viscoelastic parameters combined with other influencing parameters. Band structures of the Pb/epoxy phononic crystal in the square lattice (a) and triangular lattice (b) for different (τ=1×10-5 s) [Display omitted] [ABSTRACT FROM AUTHOR]

• Bloch waves propagation in periodic thermodiffusive layered materials is studied • Trasfer matrix method is exploited to determine material frequency band structure • Palindromic characteristic polynomial is solved in terms of Floquet multipliers • Homogeneous and inhomoegenous waves propagation are investigated • Effects of thermodiffusive coupling on elastic waves propagation are studied The dynamic behaviour of periodic thermodiffusive multi-layered media excited by harmonic oscillations is studied. In the framework of linear thermodiffusive elasticity, periodic laminates, whose elementary cell is composed by an arbitrary number of layers, are considered. The generalized Floquet-Bloch conditions are imposed, and the universal dispersion relation of the composite is obtained by means of an approach based on the formal solution for a single layer together with the transfer matrix method. The eigenvalue problem associated with the dispersion equation is solved by means of an analytical procedure based on the symplecticity properties of the transfer matrix to which corresponds a palindromic characteristic polynomial, and the frequency band structure associated to wave propagating inside the medium is finally derived. The proposed approach is tested through illustrative examples where thermodiffusive multilayered structures of interest for renewable energy devices fabrication are analyzed. The effects of thermodiffusion coupling on both the propagation and attenuation of Bloch waves in these systems are investigated in detail. [ABSTRACT FROM AUTHOR]

• The elastic wave propagation in imass embedded pre-stressed regular and re-entrant hexagonal lattices is investigated. • The Bloch theorem in combination with the finite element method is introduced to obtain the band structures of the proposed lattice systems. • Veering phenomena and zero-frequency band gaps are identified in the band strucutre diagrams. • The Hurty-Craig-Bampton model reduction method is introduced to analyze the in-plane dynamics of the embedded hexagonal structures. This paper investigates the elastic wave propagation, mode veering, and in-plane vibration of pre-stressed hexagonal lattice embedded in an elastic medium and composed of axially loaded Timoshenko beams with attached point masses. The frequency band structure of the lattice system is obtained by solving the corresponding eigenvalue problem based on the Bloch theorem and the finite element method. The parametric study is performed by investigating the effects of the pre-stress magnitude, stiffness of elastic medium, and attached point masses on the band structure of a lattice unit cell. For simulating the free vibration behavior of the proposed lattices with different topologies, the Hurty-Craig-Bampton method is introduced to reduce the number of degrees of freedom. Based on the reduced finite element model, the natural frequencies are determined for various boundary conditions. The additional interface reduction technique, called system-level reduction, has been observed to achieve accurate results compared to that of the full model. Numerical experiments demonstrated a significant influence of the additional masses, pre-stress, and stiffness of elastic medium on Bloch waves and eigenvalues of the proposed lattice systems. The effects of different parameters on the emergence of mode veering phenomenon and band gaps are investigated in detail. [ABSTRACT FROM AUTHOR]

Two-dimensional surface waves are a cornerstone for future integrated photonic circuits. They can also be beneficially exploited in sensing devices by offering dark-field illuminations of objects. One major problem in sensing schemes arises from the individual sensing objects: the interaction of surface waves with an object reduces the field amplitude, and the readout of other objects along the propagation path suffers from this reduced signal. Here we show in two experiments that nondiffracting and self-healing Bloch surface waves can be launched using a Fresnel axicon (i.e., fraxicon). First, we visualize the generation of an array of multiple focal spots by scanning near-field optical microscopy in the infrared. With a second device operating in the visible, we demonstrate the self-healing effect directly using a far-field readout method by placing metallic nanoantennas onto the multiple focal spots of the fraxicon. Our study extends the versatile illumination capabilities of surface wave systems. Two-dimensional surface waves play an important role in optical systems such as sensing devices. The authors experimentally and theoretically demonstrate a method for multiple self-healing surface wave beams which can help overcome issues related to reduction in signal strength when surface waves encounter obstacles during their propagation. [ABSTRACT FROM AUTHOR]