Bloch Floquet Research Articles

Functionally Graded Soft Dielectric Elastomer Phononic Crystals: Finite Deformation, Electro-Elastic Longitudinal Waves, and Band Gaps Tunability via Electro-Mechanical Loading

Active tuning of elastic wave band gaps received significant attention in the recent past. To this end, soft dielectric elastomers (DEs) have indicated great promise owing to their capability to undergo large reversible deformations and deformation-dependent constitutive properties. In this paper, we investigate the longitudinal wave band gaps an infinite periodic soft functionally graded dielectric elastomer subjected to electro-mechanical biasing fields. Material properties of the functionally graded DE are described by the power law. The nonlinear hyperelastic compressible neo–Hookean model is employed for characterizing the constitutive behavior of the soft functionally graded dielectric elastomer. The finite element method with Bloch–Floquet theory is employed for finding the band-structure of the infinitely periodic soft functionally graded DE structure. A parametric study brings out the effect of biasing fields and power law exponent on the band gap characteristics. The biasing mechanical prestress, the electric displacement field show significant influence on position and the width of band gaps. The inferences reported here can find their potential use in the design of soft DE wave devices with tunable band structures.

Enhanced attenuation band in active 2D mass-in-mass metamaterial using static output feedback

Active feedback control is implemented between the inner and outer masses of a 2D mass-in-mass metamaterial to widen the attenuation bandwidth by reducing the displacement of the outer mass of each unit cell. The optimal static output feedback control is applied to a single representative unit. The gain is computed from the analysis of a single unit cell and further used along with the Bloch–Floquet theorem for obtaining the dispersion relationship for an infinite 2D discrete square lattice metamaterial. The proposed model is validated by comparing the results against the finite element simulation in ANSYS. The control gain couples the displacements and velocities, which generates the damping phenomenon even in an undamped system. The controller design requires feedback of only a few measured output instead of all the state variables, and offline computation of the feedback gains reduces the time delay. Effective mass analysis and energy dissipation mechanism are performed which sheds light on the physics underlying the attenuation phenomena. An ultra-wide uninterrupted attenuation band can be perceived as the negative effective mass region significantly extends beyond the standard limit by employing the active feedback control. Additionally, the attenuation bandwidth exists in the low frequency due to the damping induced from the active control.

Interfacial delamination-induced unidirectional propagation of guided waves in multilayered media

Unidirectional nonreciprocal wave propagation is an unprecedented phenomenon, which has attracted much research interest. Connecting a phononic crystal with an asymmetric structure to break the spatial inversion symmetry is a popular manner to realize this phenomenon using the wave mode transformation. In this paper, a new model is proposed based on a single periodic structure. The arrays of asymmetric and symmetric interfacial delaminations are intentionally introduced into the top and the bottom part of a stack of periodic elastic layers, respectively. So, the structural spatial inversion symmetry can be broken and the guided waves can pass through the whole structure only from the top side with the changed mode generated by the array of asymmetric interfacial delaminations. Thus, it is indispensable for the part of phononic crystal that the partial band-gaps of symmetric and antisymmetric guided waves have to be separated, which is the reason why we introduce the array of symmetric central or side interfacial delaminations into the stack of periodic elastic layers. The transmission spectra of the guided waves and the dispersion curves for the unit cell imposed by the Bloch–Floquet boundary condition are both calculated by the spectral element method. Then, the interfacial delamination-induced unidirectional propagation of guided waves in the finite stack of periodic elastic layers is numerically confirmed. This paper provides a new concept to control the waves propagating in phononic crystals via the insertion of some interfacial delaminations or cracks.

Bandgap merging with double-negative metabeam

We theoretically investigate the bandgap formation in an Euler–Bernoulli beam-based metastructure with periodically attached double negative 3-dof local resonators. The idiosyncratic physical phenomena found in mechanical metamaterials, such as negative stiffness and mass, are introduced in the attached local resonator to design the double negative metabeam. The transfer matrix method with the Bloch–Floquet formulation is implemented in a unit cell of the double negative metabeam. The complex band structure describes the existence of various locally resonant attenuation bandgaps on the frequency spectrum. Further system analysis is conducted to illustrate the merging of the negative stiffness and mass controlled locally resonant bandgaps. A significant increase of 164% is found in the attenuation bandwidth due to the merging of locally resonant bandgaps.

Topology optimization of soft compressible phononic laminates for widening the mechanically tunable band gaps

Active tuning of frequency bandgaps has received significant attention in the recent past. To this end, soft compressible phononic composites have indicated great promise owing to their capability to undergo large reversible deformations and deformation dependent constitutive properties. In this investigation, we present an optimization framework for arriving at the optimum topology of a two-phase soft compressible phononic composite for maximizing the frequency gap between the first four consecutive frequency bands of longitudinal wave propagation. The laminate is subjected simultaneously to an equal biaxial pre-stretch in the plane perpendicular to the direction of wave propagation and a prestress in the direction of wave propagation. The nonlinear hyperelastic neo-Hookean material model is employed for characterizing the constitutive behavior of the soft compressible laminate phases. The finite element method with Bloch–Floquet theory is employed for finding the band-structure of the infinitely periodic soft laminate. A gradient-based method of moving asymptotes is used to solve the underlying topology optimization problem. In particular, we present the optimum unit cell designs for two separate problems involving maximization of the absolute width of the band gap and that of the relative width of the band gap when compared to that in the un-deformed laminate. Test cases are presented to highlight the effects of the levels of biaxial prestretch and the longitudinal prestress. Our results indicate that a compressional longitudinal prestress and a lateral prestretch have a favorable impact on widening of the frequency band-gaps. The numerical framework and the inferences reported here can find their potential use in the optimal design of soft compressible composites used in acoustic applications.

Spectral-Element Spectral-Integral (SESI) Method for the 1-D Bloch (Floquet) Periodic Problems With Scatterers Embedded in Multiple Regions of 2-D Layered Media

The hybrid spectral-element spectral-integral (SESI) method is developed for the electromagnetic (EM) scattering in 1-D Bloch (Floquet) periodic problems with scatterers embedded in multiple regions of 2-D layered media. The periodic layered medium Green’s function (PLMGF) is derived for the SESI method so that it can simulate EM scattering by horizontally periodic scatterers placed in an arbitrary number of layers. The SESI method is a combination of the spectral-element method (SEM) and the spectral-integral method (SIM). The SEM is applied only in regions with scatterers and has the merits of exponential convergence at a low spatial sampling density, while the SIM serves as an exact radiation boundary condition for the bottom and top boundaries of each SEM region. Therefore, there is no need to discretize the regions with pure layered media; furthermore, the inverses of these SIM matrices are given exactly and explicitly with only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N\log N)$ </tex-math></inline-formula> CPU time. Three numerical experiments are performed to show the accuracy and efficiency of the SESI method.

Influence of attached inertia and resonator on the free wave propagation in 2D square frame grid lattice metamaterial

This paper presents spectral element based approach for determining free planner wave propagation in two-dimensional periodic square-frame-grid-lattices (SqL). Implementing the newly developed nonlinear eigenvalue solver in conjunction with Bloch–Floquet periodic boundary condition, band structures for various configurations of SqL with attached mass or spring-mass resonator are analysed. Variation of the attenuation bandwidth is studied for different mass and frequencies of the attachments to the SqL. At the natural frequency of the resonator, an attenuation bandgap can be perceived; therefore, by tuning the natural frequency of the resonator, sub-wavelength band-gap can be obtained at the desired frequency for enhanced vibration or noise isolation. However, in few cases, mostly while the resonators are attached at the center point of the SqL, the attenuation bandgap near the natural frequency of the resonator may disappear as the center point lying in the node for that free wave frequency. Additionally, existence of dirac cone, eigenfrequency-loci veering are observed in the band-structures. The iso-frequency contours are also investigated to develop an insight comprehension about the underlying physics of energy transmission and mechanics of wave propagation. Various salient wave propagation features, namely self-collimation, lensing, and negative refraction of group velocity are identified and elucidated in this paper.

Light-induced emergent phenomena in 2D materials and topological materials

Light-matter interaction in 2D and topological materials provides a fascinating control knob for inducing emergent, non-equilibrium properties and achieving new functionalities in the ultrafast time scale (from fs to ps). Over the past decade, intriguing light-induced phenomena, e.g., Bloch-Floquet states and photo-induced phase transitions, have been reported experimentally, but many still await experimental realization. In this Review, we discuss recent progress on the light-induced phenomena, in which the light field could act as a time-periodic field to drive Floquet states, induce structural and topological phase transitions in quantum materials, couple with spin and various pseudospins, and induce nonlinear optical responses that are affected by the geometric phase. Perspectives on the opportunities of proposed light-induced phenomena as well as open experimental challenges are also discussed.

Possible Routes to Obtain Enhanced Magnetoresistance in a Driven Quantum Heterostructure with a Quasi-Periodic Spacer.

In this work, we perform a numerical study of magnetoresistance in a one-dimensional quantum heterostructure, where the change in electrical resistance is measured between parallel and antiparallel configurations of magnetic layers. This layered structure also incorporates a non-magnetic spacer, subjected to quasi-periodic potentials, which is centrally clamped between two ferromagnetic layers. The efficiency of the magnetoresistance is further tuned by injecting unpolarized light on top of the two sided magnetic layers. Modulating the characteristic properties of different layers, the value of magnetoresistance can be enhanced significantly. The site energies of the spacer is modified through the well-known Aubry–André and Harper (AAH) potential, and the hopping parameter of magnetic layers is renormalized due to light irradiation. We describe the Hamiltonian of the layered structure within a tight-binding (TB) framework and investigate the transport properties through this nanojunction following Green’s function formalism. The Floquet–Bloch (FB) anstaz within the minimal coupling scheme is introduced to incorporate the effect of light irradiation in TB Hamiltonian. Several interesting features of magnetotransport properties are represented considering the interplay between cosine modulated site energies of the central region and the hopping integral of the magnetic regions that are subjected to light irradiation. Finally, the effect of temperature on magnetoresistance is also investigated to make the model more realistic and suitable for device designing. Our analysis is purely a numerical one, and it leads to some fundamental prescriptions of obtaining enhanced magnetoresistance in multilayered systems.

Trapped Modes and Negative Refraction in a Locally Resonant Metamaterial: Transient Insights into Manufacturing Bounds for Ultrasonic Applications

The transient scattering of in-plane elastic waves from a finite-sized periodic structure, comprising a regular grid of Swiss-cross holes arranged according to a square lattice, is considered. The theoretical and numerical modelling focuses on the unexplored ultrasonic frequency regime, well beyond the first, wide, locally resonant band-gap of the structure. Dispersive properties of the periodic array, determined by Bloch–Floquet analysis, are used to identify candidates for high-fidelity GPU-accelerated transient scattering simulations. Several unusual wave phenomena are identified from the simulations, including negative refraction, focusing, partial cloaking, and wave trapping. The transient finite element modelling framework offers insights on the lifetimes of such phenomena for potential practical applications. In addition, nonideal counterparts with rough edges are modelled using characteristic statistical parameters commonly observed in additive manufacturing. The analysis shows that the identified wave effects appear likely to be robust with respect to potential manufacturing uncertainties in future studies.

Ultra-wide bandgap in active metamaterial from feedback control

To widen the attenuation bandwidth in a mass-in-mass metamaterial chain, active feedback control is employed within the unit cell. The optimal control via Linear Quadratic Regulator is used to ensure the stability of the solution as well as adaptive tuning of the system. The key concept is to obtain a wider bandwidth by reducing the displacement amplitude of the outer mass of each unit of the metamaterial. Transmittance for finite number of units and dispersion diagram for infinitely long metamaterial chain are obtained using the backward-substitution method and Bloch–Floquet’s theorem, respectively. Results evidenced that an ultra-wide uninterrupted attenuation band starting from very low free wave frequency can be obtained. The dispersion diagram primarily depends on the frequency ratio of the resonating unit with the external main unit. The effect of the mass ratio is insignificant while active feedback control is implemented. This unique phenomenon implies that for a very low mass ratio also a wider bandwidth can be achieved. This enables to design a lightweight structure having wider bandwidth in low-frequency regime.

Lateral flexural vibration reduction in a periodic piping system enhanced with two-degrees-of-freedom resonators

Pipe systems are commonly used in the process and power industries to transport fluid from one terminal to others. Propagation behaviour of lateral flexural waves in a pipe coupled with periodic rack structure is investigated. The pipe-rack system considered in this study is a practical case and is realized as a pipe on periodic elastic supports, while a pipe on simple and without supports represents special cases when the rack stiffness leads to extreme values. The propagation constant relations in terms of frequency are derived using Bloch–Floquet theorem which are successively verified with finite element models. The results show that a pipe with rack creates a narrow locally resonant band gap in low-frequency range which is caused by the first natural mode of the rack. Conversely, a pipe on simple supports entails only Bragg-type band gaps, while a pipe without supports carries no band gap. For tuning the band gap properties, a two-degrees-of-freedom lateral localized resonator is attached to the centre of each unit cell of the pipe. It is found that certain frequency ranges in the targeted pass bands are effectively controlled by the resonator. Furthermore, the effect of various resonator parameters, i.e. mass ratio, stiffness and damping, on band gaps is examined. It is observed that the band gaps are vanished when damping is introduced in the system. The results show a promising way to flexural vibration control of a periodic piping system with various boundary conditions.

Modelling wave dispersion in fluid saturating periodic scaffolds

Acoustic waves in a slightly compressible fluid saturating porous periodic structure are studied using two complementary approaches: 1) the periodic homogenization (PH) method provides effective model equations for a general dynamic problem imposed in a bounded medium, 2) harmonic acoustic waves are studied in an infinite medium using the Floquet-Bloch (FB) wave decomposition. In contrast with usual simplifications, the advection phenomenon of the Navier–Stokes equations is accounted for. For this, an acoustic approximation is applied to linearize the advection term. The homogenization results are based the periodic unfolding method combined with the asymptotic expansion technique providing a straight upscaling procedure which leads to the macroscopic model defined in terms of the effective model parameters. These are computed using the characteristic responses of the porous microstructure. Using the FB theory, we derive dispersion equations for the scaffolds saturated by the inviscid, or the viscous, barotropic fluids, whereby the advection due to a permanent flow in the porous structures is respected. A computational study is performed for the numerical models obtained using the finite element discretization. For the FB methods-based dispersion analysis, quadratic eigenvalue problems must be solved. The numerical examples show influences of the microstructure size and of the advection generating an anisotropy of the acoustic waves dispersion.

Flexural Waves in Elastically Coupled Telescopic Metabeams

Abstract This paper investigates the flexural wave propagation through elastically coupled telescopic metabeams. It is assumed that the metabeam is formed by connecting successive beams with each other using distributed elastic springs. The equations of motion of a representative unit of the above-mentioned novel structural form are established by dividing it into three constitutive components that are two side beams, modeled employing the Euler–Bernoulli beam equation and an elastically coupled articulated distributed spring connection (ECADSC) at middle. ECADSC is modeled as parallel double beams connected by distributed springs. The underlying mechanics of this system in context of elastic wave propagation is unique when compared with the existing state of art in which local resonators, inertial amplifiers, etc. are attached to the beam to widen the attenuation bandwidth. The dynamic stiffness matrix is employed in conjunction with Bloch–Floquet theorem to derive the band structure of the system. It is identified that the coupling coefficient of the distributed spring layer and length ratio between the side beams and the elastic coupling plays the key role in the wave attenuation. It has been perceived that a considerable widening of the attenuation bandgap in the low frequency can be achieved while the elastically distributed springs are weak and distributed in a small stretch. Specifically, 140% normalized bandgap can be obtained only by tuning the stiffness and the length ratio without adding any added masses or resonators to the structure.

Fully decoupling geometry from discretization in the Bloch–Floquet finite element analysis of phononic crystals

An immersed enriched finite element method is proposed for the analysis of phononic crystals (PnCs) with finite element (FE) meshes that are completely decoupled from geometry. Particularly, a technique is proposed to prescribe Bloch–Floquet periodic boundary conditions strongly on non-matching edges of the periodic unit cell (PUC). The enriched finite element formulation effectively transforms a periodic non-confirming discretization into an enriched node-to-node periodic discretizations where periodicity is enforced by any standard procedure. The enriched formulation is also used to describe the interior material interface. This completely eliminates the tedious process of generating matching or fitted meshes during the design process, as it allows changing the inclusion’s geometry as well as the PnC’s lattice type without changing the FE mesh. The proposed approach, which is used to analyze phononic crystals in 1-D, 2-D, and 3-D using structured meshes, exhibits the same performance as the standard finite element analysis on fitted meshes.

Rapid detection of cracks in the rail foot by ultrasonic B-scan imaging using a shear horizontal guided wave electromagnetic acoustic transducer

Transverse cracks in the rail foot are responsible for the breaking of the rail. The existing non-destructive testing technology for high-speed railway track is difficult to perform in-situ and rapid detection for the rail foot, which poses a hidden danger to driving safety. In this study, a new method is proposed for the in-situ and rapid detection of cracks in the rail foot by ultrasonic B-scan imaging. A finite element method (FEM) based on the Bloch–Floquet boundary and domain constraint were employed to calculate the dispersion curves of shear horizontal (SH)-like guided waves propagating in the rail foot. SH guided waves mainly vibrate with in-plane displacement, and they are less affected by the rail components such as elastic clips and pads. Based on the calculated dispersion curves, an SH-guided wave electromagnetic acoustic transducer (EMAT) with a center frequency of 0.154 MHz was developed, and ultrasonic B-scan imaging was performed to detect straight cracks in the rail foot. Besides, the synchrosqueezed wavelet transform (SWT) method was proposed to remove noises and higher-order guided wave modes from the original ultrasonically detected signals. Results show that FEM with the Bloch–Floquet boundary and domain constraint can be used to solve the dispersion curves of the SH-like guided waves of the rail foot accurately and easily. The application of SWT can increase the lift-off of the EMAT, eliminate the noise, remove the higher-order SH-like guided wave modes, improve the detection efficiency, and enhance the quality of ultrasonic B-scan images. After the SWT processing, the signal-to-noise ratio (SNR) of the ultrasonic signal is increased by at least 5.98 dB. When the lift-off of the EMAT is 4 mm and without synchronous averaging, the on-line and rapid ultrasonic B-scan imaging of the rail foot using guided wave EMAT can still achieve a reasonable SNR.

A new prescription to achieve a high degree of spin polarization in a spin–orbit coupled quantum ring: efficient engineering by irradiation

The present work discusses the possibility to achieve a high degree of spin polarization in a three-terminal quantum system. Irradiating the system, subjected to Rashba spin–orbit (SO) interaction, we find high degree of spin polarization under a suitable input condition along with different magnitudes and phases at the two output leads. The system is described within a tight-binding (TB) framework and the effect of irradiation is incorporated following the Floquet–Bloch (FB) ansatz. All the spin-dependent transmission probabilities are evaluated through Green’s function technique using Landauer–Büttiker formalism. Several possible aspects are included to make the system more realistic and examined rigorously in the present work. To name a few, the effects of irradiation, SO interaction, interface sensitivity, system size, system temperature are investigated, and finally, the role of correlated impurities are studied. Despite having numerous proposals available to generate and manipulate spin-selective transmissions, such a prescription exploiting the irradiation effect is relatively new to the best of our concern.

Mixed Spectral-Element Methods for 3-D Maxwell’s Eigenvalue Problems With Bloch Periodic and Open Resonators

Two mixed spectral-element methods (MSEMs) are proposed to solve Maxwell’s eigenvalue problems with Bloch (Floquet) periodic and open resonators to remove the dc spurious modes present in the traditional numerical methods. The first MSEM combines the variational form of Gauss’s law to make the electric field satisfy the divergence-free condition. This method does not need to modify the Arnoldi algorithm for the eigenvalue problems, but the number of degrees of freedom (DOFs) is larger than that of the traditional spectral-element method (SEM). The second MSEM, as a generalization of the tree–cotree technique, can eliminate the dc spurious modes by imposing the constrained equations on the Ritz vectors in the Arnoldi algorithm without increasing the DOF from the traditional SEM. We expand the electric field in terms of the edge basis functions associated with the cotree edges and the gradient of the nodal basis functions so that the discrete gradient matrix $\bar {\bar {\mathrm {G}}}$ consists of the identity matrix $\bar {\bar {\mathrm {G}}}_{t}$ and the zero matrix $\bar {\bar {\mathrm {G}}}_{c}$ , making the matrix inversion of $\bar {\bar {\mathrm {G}}}_{t}$ trivial. Finally, in the numerical experiments, we compare the computational costs of the MSEMs and finite-element method (FEM) in COMSOL to illustrate the high efficiency of the MSEMs.

Multiscale stability analysis of periodic magnetorheological elastomers

We analyze instability phenomena of periodic magnetorheological elastomers in the framework of computational homogenization. Our focus is on two kinds of instabilities given by macroscopic material and microscopic structural instabilities. While the first are related to the rank-one convexity of an associated homogenized energy density, the latter are related to the coercivity of an associated microscopic boundary value problem. At both scales we detect instabilities of equilibrium states by superimposing wave-like perturbations. At macroscopic scale we consider classical plane waves giving rise to the definition of a generalized acoustic tensor. At microscopic scale we exploit Bloch–Floquet wave analysis, which allows to determine critical buckling modes of microstructures based on computations at unit-cell level. The microscopic material response is governed by a four-field variational principle of magneto-elasticity that is embedded in a framework of first-order computational homogenization. A series of numerical simulations reveals a spectrum of complex pattern transformations that could be triggered by appropriate microstructure design and coupled magneto-mechanical loading.

Dual-Polarized All-Metallic Metagratings For Perfect Anomalous Reflection

We theoretically formulate and experimentally demonstrate the design of metagratings (MGs) composed of periodic rectangular grooves in a metallic medium, intended for perfect anomalous reflection. Using mode matching, a semianalytical scheme for analysis and synthesis of such MGs, containing multiple, arbitrarily arranged, grooves per period, is derived. Following the typical MG design approach, we use this formalism to identify the relevant Floquet-Bloch (FB) modes and conveniently formulate constraints for suppression of spurious scattering, directly tying the structure's geometrical degrees of freedom (DOFs) to the desired functionality. Solving this set of constraints, in turn, yields a detailed fabrication-ready MG design, without any full-wave optimization. Besides providing means to realize highly-efficient beam deflection with all-metallic formations, we show that the rectangular (two-dimensional) groove configuration enables \emph{simultaneous} manipulation of both transverse electric (TE) and transverse magnetic (TM) polarized fields, unavailable to date with common, printed-circuit-board-based, microwave MGs. In addition, we highlight a physical limitation on the TE-polarization performance, preventing the ability to achieve perfect anomalous reflection in any desired angle. These capabilities are verified using three MG prototypes, produced with standard computer numerical control (CNC) machines, demonstrating both single- and dual-polarized control of multiple diffraction modes. These results enable the use of MGs for a broader range of applications, where dual-polarized control is required, or all-metallic devices are preferable (e.g., spaceborne systems or at high operating frequencies).