Bloch-Floquet Theory Research Articles

Bandgap structure of tensegrity mass–spring chains equipped with internal resonators

This work studies the dispersion relation of a Maxwell type mass–spring chain formed by lumped masses and the parallel arrangement of two different types of tensegrity prisms. Use is made of the Bloch–Floquet theory of discrete systems in association with a linearized model of the response of the tensegrity units under compression loading. Such a modeling is aimed at studying the propagation of compression waves under small perturbations of the initial equilibrium state of the system. For a given value of the cable’s prestress, the tensegrity systems connecting the lumped masses react as elastic springs, which exhibit axial deformations accompanied by relative twisting rotations of the terminal bases. The twisting motion of the chain affects the expression of the kinetic energy, and is accounted for by introducing a suitable definition of equivalent masses. The bandgap structure of the analyzed system is analytically determined and numerical results are obtained for a chain formed by physical models tensegrity θ=1 prisms aligned in parallel with minimal tensegrity prisms. The given results highlight the highly tunable frequency bandgap properties of tensegrity mass–spring chains exhibiting internal resonance capabilities.

Forced vibrations of a finite length metabeam with periodically arranged internal hinges and external supports

The paper investigates a problem of forced vibrations of two finite length metabeams: one is with periodically arranged internal hinges and the second one with periodically arranged internal hinges and external supports. Based on the Euler–Bernoulli beam theory and the transfer matrix method, general solutions of the periodic beams are obtained. Applying the Bloch-Floquet theory, the explicit expressions are derived defining the metabeams band gap structures. The corresponding band gap dispersion curves are plotted and analysed. It is shown that when the frequency of forced vibrations coincides with the band gap frequencies a strong localization of flexible waves occurs at the interfaces of the periodic beams. The localization of flexible waves increases significantly with the number of hinges and supports.

Modified non-dominated sorting genetic algorithm-II for the optimal design of soil-concrete periodic plane wave barriers

This paper uses the Bloch-Floquet theory based on the modified non-dominated sorting genetic algorithm-II (NSGA-II) to investigate the optimal design of wave barriers to attenuate plane waves. The primary aim is to find the optimal design of the plane wave barriers with a wide bandgap at a low-frequency range having low construction cost. Consequently, a modification of NSGA-II has been developed and adopted to obtain the optimal arrangement of concrete within different 8 × 8, 10 × 10, and 12 × 12 grid sizes in the wave barrier unit cell by considering three objective functions as (1) maximize bandgap width (BW), (2) minimize average bandgap frequency (ABF), and (3) minimize filling ratio (FR). The Pareto fronts represent a wide range of objective functions in the problem space, making the proposed method effective for designing plane wave barriers. The performance of the optimal barriers is examined with the three-dimensional finite element (FE) simulation of the finite lattice in the frequency domain. The finite lattice results confirm the obtained bandgaps for the optimal unit cells and verify their efficacy in attenuating plane S-waves. Furthermore, the effects of the number of barriers and the barrier length are studied as two effective parameters affecting plane S-wave attenuation.

Geometric Aspects of Nonlinear and Nonequilibrium Phenomena

We review recent developments in the research of nonlinear and nonequilibrium phenomena in solids focusing on their geometrical aspects. We start with introducing the basic concepts of geometrical phases of Bloch electrons and Floquet theory for periodically driven systems. Then we review recent attempts to engineer topological phases in nonequilibrium matters such as graphene, magnets, and superconductors irradiated with circularly polarized light. We next review a bulk photovoltaic effect of inversion broken materials focusing on the shift current response. The shift current is described with Berry connections and has a close relationship to the modern theory of polarization. We further review recent extensions of the shift current to correlated electron systems. Finally, we explain the geometric diabatic time evolution in the Zener tunneling process and its consequences on nonreciprocal transport.

Cascaded parametric amplification based on spatiotemporal modulations

Active devices have drawn considerable attention owing to their powerful capabilities to manipulate electromagnetic waves. Fast and periodic modulation of material properties is one of the key obstacles to the practical implementation of active metamaterials and metasurfaces. In this study, to circumvent this limitation, we employ a cascaded phase-matching mechanism to amplify signals through spatiotemporal modulation of permittivity. Our results show that the energy of the amplified fundamental mode can be efficiently transferred to that of the high harmonic components if the spatiotemporal modulation travels at the same speed as the signals. This outstanding benefit enables a low-frequency pump to excite parametric amplification. The realization of cascaded parametric amplification is demonstrated by finite-difference time-domain (FDTD) simulations and analytical calculations based on the Bloch–Floquet theory. We find that the same lasing state can always be excited by an incidence at different harmonic frequencies. The spectral and temporal responses of the space-time modulated slab strongly depend on the modulation length, modulation strength, and modulation velocity. Furthermore, the cascaded parametric oscillators composed of a cavity formed by photonic crystals are presented. The lasing threshold is significantly reduced by the cavity resonance. Finally, the excitation of cascaded parametric amplification relying on the Si-waveguide platform is demonstrated. We believed that the proposed mechanism provides a promising opportunity for the practical implementation of intense amplification and coherent radiation based on active metamaterials.

Microstructural design for elastic wave attenuation in 3D printed nacre-like bioinspired metamaterials lightened with hollow platelets

In this work, the wave propagation properties of lightened bioinspired composite materials are investigated. A new lightened nacre-like microstructure is proposed demonstrating that the incorporation of hollow platelets inside the soft matrix and the application of specific prestress states represent a viable way for tuning the complete band gaps range without varying the geometry length scale and also improving the mechanical properties in terms of homogenized tangent moduli. The wave dispersion relations are obtained by adopting the Bloch-Floquet theory to investigate the appearance of complete band gaps for several geometrical and material parameters (platelets aspect ratio, matrix joints thickness, void volume fraction, and shear modulus contrast). The obtained results reveal new combinations of geometrical and material parameters for which excellent wave absorption capabilities in nacre-like periodic composite microstructure are obtained thanks to the application of prestress states which may provide new opportunities in the microstructural design of 3D printed composite metamaterials.

Plane waves mitigation in saturated porous media using optimal periodic wave barriers

Ground vibrations induced by underground transportation, human activities, and earthquakes can damage civil infrastructures. The damage can be more pronounced in saturated soils common in many parts of the world. This paper investigates the optimal design of wave barriers in the fluid-saturated porous media to mitigate body waves using the Bloch-Floquet theory. The primary purpose is to find the configuration of the optimal barriers that provide wide bandgaps for plane waves. Two objective functions are considered by adopting the Genetic Algorithm (GA): maximizing the bandgap width and the relative bandgap at each filling fraction (FR) in a unit cell representing the infinite lattice. The optimal barrier configuration obtained from infinite periodic lattice is subsequently arranged in a more practical finite lattice arrangement. The examination of the optimal finite lattice in both time and frequency domains confirms the obtained bandgaps for the infinite lattice and demonstrates their efficacies in attenuating damaging ground vibrations. Especially, the time domain analyses show that the optimal configurations perform better for S-waves than the P-waves.

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.

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.

A dual-band, dual-polarized, CPW-fed wideband antenna loaded with via less CRLH-MTM TL for 5G mm-Wave communication

This paper effectively implements the process of bandwidth enhancement and realization of circular polarization (CP) of metamaterial-based antenna by loading composite right/left-handed (CRLH) transmission line (TL) unit cell without vias. This antenna is a symmetrical structure that comprises of square slot, two asymmetric meander lines (AML) in y-direction, an interdigital capacitance (IDC) in x-direction, inductive rhombus (IR), and coplanar waveguide (CPW) feed line in a single plane. Mender lines (ML) provides the needed shunt inductance whereas IDC in the signal line provides series capacitance for left-handed TL. The proposed metamaterial-based antenna structure essentially produces three resonating bands. The simulated first band has a frequency range of 14.5 GHz to 15.4 GHz due to zeroth-order resonance (ZOR), which can be used for satellite communication. Dispersion analysis is done to verify metamaterial (MTM) behavior at ZOR frequency using Bloch-Floquet theory. The second and third resonance are due to coupling capacitances, created in antenna structure, which are merged to make a broadband antenna in millimeter-Wave (mm-Wave) range from 28 GHz to 39.2 GHz, for 5G cellular communication by increasing the diagonal length of IR. The CP is attained by converting symmetric meander lines (MLs) to asymmetric MLs to excite the orthogonal field components of equal amplitude. The simulated results are compared to measured results and found to be in a good match. The measured impedance bandwidth (IBW) for the Ku-band is 7.31% (14.5–15.6) GHz with linear polarization (LP) characteristics while 31% (28–38.2) GHz in mm-Wave (Ka-band) range. The measured axial ratio bandwidth (ARBW) is 5.2 GHz (13.9%, 33.6 GHz-38.8 GHz) in mm-Wave range. The proposed antenna has a peak gain of 6.7 dBi and maximum efficiency is 93%. The overall dimension of the presented antenna is 0.5λ0 × 0.5λ0 × 0.025λ0 at 14.9 GHz.

Shielding performance of T-shaped periodic barrier for surface waves in transversely isotropic soil

Aiming at the vibration isolation in transversely isotropic soil, a T-shaped partially embedded periodic barrier for surface waves is proposed, and its shielding performance is explored by using finite element method combined with Bloch-Floquet theory. Seven independent dimensionless material parameters are derived and their influences on band gaps are discussed numerically. The results show that the band gaps exhibit strong sensitivity to the three parameters out of seven, and the band gaps are far wider in transversely isotropic soils than that in the isotropic. The mass density ratio and the shear modulus ratio of the barrier to the soil, as well as the length ratio of the barrier above the ground to that below, can be used to adjust band gaps effectively to meet the shielding requirements for different frequency ranges under different anisotropic soils. As a case of study, the El Centro seismic wave is considered and found that it can be considerably attenuated by the designed periodic barrier.

Validation of FEM models based on Carrera Unified Formulation for the parametric characterization of composite metamaterials

This work focuses on the study of composite metamaterials to be employed as possible lightweight insulation systems for noise and vibrations. In particular, the dispersion relations are derived by applying the Bloch-Floquet theory to the unit cell of the periodic microstructure. Advanced beam finite elements based on Carrera Unified Formulation are here extended, for the first time, to the dynamic characterization of these materials. Moreover, transmission curves are computed to validate the band gaps encountered in the analysis of dispersion behavior. The finite element model is first assessed by evaluating the dispersion behavior of some metamaterial configurations proposed in the literature. Finally, the model is applied to the parametric characterization of a composite metamaterial made of a melamine foam matrix and periodic distribution of cylindrical aluminium inclusions. The results show that it is possible to tune the band gaps of the metamaterial to lower frequencies by simply varying the dimensions of the unit cell and keeping constant its equivalent density.

Preliminary study on the periodic base isolation effectiveness and experimental validation

Periodic material possesses distinct characteristics that can block some certain frequency waves propagate through the structure, which can be used to reduce the unwanted seismic waves in civil engineering. Base on the Bloch-Floquet theory, the certain frequencies, terms frequency band gaps, are theoretically presented and derived. A numerical model of 1D periodic foundation is established to investigate the filtering effects of elastic waves. The dynamic response and filtering effects of a three-storey steel frame with a 1D periodic foundation and concrete foundation are performed and compared through the numerical method and shaking table tests. Ambient vibration, earthquake ground motions, and sine waves are applied as the input waves. The results show that the seismic waves cannot propagate in the 1D periodic foundation when the frequencies of the waves fall into the frequency band gaps of the foundation. Simultaneously, a single unit cell has proven to possess the same frequency band gaps as those with an infinite number of unit cells and has enough response reduction inside the frequency band gaps of the periodic foundation. Consequently, the proposed periodic foundation can be used to apply in isolating environmental vibration or seismic waves in civil engineering through adjusting the frequency band gaps enveloping the exciting frequencies of the waves.

Surface-wave-assisted nonreciprocity in spatio-temporally modulated metasurfaces

Emerging photonic functionalities are mostly governed by the fundamental principle of Lorentz reciprocity. Lifting the constraints imposed by this principle could circumvent deleterious effects that limit the performance of photonic systems. Most efforts to date have been limited to waveguide platforms. Here, we propose and experimentally demonstrate a spatio-temporally modulated metasurface capable of complete violation of Lorentz reciprocity by reflecting an incident beam into far-field radiation in forward scattering, but into near-field surface waves in reverse scattering. These observations are shown both in nonreciprocal beam steering and nonreciprocal focusing. We also demonstrate nonreciprocal behavior of propagative-only waves in the frequency- and momentum-domains, and simultaneously in both. We develop a generalized Bloch-Floquet theory which offers physical insights into Lorentz nonreciprocity for arbitrary spatial phase gradients, and its predictions are in excellent agreement with experiments. Our work opens exciting opportunities in applications where free-space nonreciprocal wave propagation is desired.

Homogenization Approach and Bloch-Floquet Theory for Band-Gap Prediction in 2D Locally Resonant Metamaterials

This paper provides a detailed comparison of the two-scale homogenization method and of the Bloch-Floquet theory for the determination of band-gaps in locally resonant metamaterials. A medium composed by a stiff matrix with soft inclusions with 2D periodicity is considered and the equivalent mass density of the homogenized medium is explicitly obtained both for in-plane and out-of-plane wave propagation through two-scale asymptotic expansion. The intervals of frequency where the effective mass is negative identify the band-gaps of the material. The Bloch-Floquet problem is then considered and, through an asymptotic analysis, its is shown that it leads to the same prediction of the band-gaps. The results are confirmed by some examples and the limits of the asymptotic approach are explicitly given and numerically verified.

Shear horizontal wave propagation along a periodic metal grating surface of a magneto-electro-elastic substrate

This paper analyzes shear horizontal (SH) waves propagating in thin metal strips deposited periodically onto the surface of a magneto-electro-elastic (MEE) substrate. The MEE substrate is assumed to be a transversely isotropic crystal with 6 mm symmetry or a hexagonal (6 mm) crystal. A field analysis method, which is based on the Bloch–Floquet theory and the coupled equations of wave motion, is employed to derive the dispersion equations and wave mode shapes. Numerical examples are presented for an MEE composite (PZT4/CoFe2O4) and an aluminum (Al) strip. For comparison, different constituent ratios of PZT4/CoFe2O4 are considered. The effect of the ratios of strip height and width to grating periodicity on frequencies, phase velocities, as well as wave mode shapes is discussed in detail. The results show that there is a bandgap at the resonance condition when the grating periodicity matches the wavelength of the SH surface wave. This metal grating surface can trap SH surface waves by slowing the SH waves. The wave mode shapes indicate that the SH surface waves are trapped mainly on these metal strips. As the ratio of strip height to grating periodicity increases, the trapping effect is more pronounced and the gap becomes wider. The ratio of strip width to grating periodicity significantly affects the behavior of SH surface waves. The present results are relevant in the application of MEE surface acoustic wave resonators.

Tuning frequency band gaps of tensegrity mass-spring chains with local and global prestress

This work studies the acoustic band structure of tensegrity mass-spring chains, and the possibility to tune the dispersion relation of such systems by suitably varying local and global prestress variables. Building on established results of the Bloch–Floquet theory, the paper first investigates the linearized response of chains composed of tensegrity units and lumped masses, which undergo small oscillations around an initial equilibrium state. The stiffness of the units in such a state varies with an internal self-stress induced by prestretching the cables forming the tensegrity units, and the global prestress induced by the application of compression forces to the terminal bases. The given results show that frequency band gaps of monoatomic and biatomic chains can be effectively altered by the fine tuning of local and global prestress parameters, while keeping material properties unchanged. Numerical results on the wave dynamics of chains under moderately large displacements confirm the presence of frequency band gaps of the examined systems in the elastically hardening regime. Novel engineering uses of the examined systems are discussed.

Controlling Nonreciprocity Using Enhanced Brillouin Scattering

The properties of space-time modulated media operating in the sub-sonic regime are discussed based on rigorous Bloch-Floquet theory. A geometrical description in the frequency-wavenumber plane is developed to provide insight into the possible interactions and their nature. It is shown that the secular equation has a singularity, which results in a weak/passive second harmonic generation process. Additionally bandgaps arising from the strong/active parametric interaction between an incident wave and its space-time harmonic, result in an inelastic Brillouin like scattering process. Hence when the incident frequency is inside a forward (backward) bandgap, a Stokes' (Anti-Stokes') scattered wave bounces back to the source. Although the forward and backward bandgaps do not generally occur at the same frequency bands, the insertion loss and gap width are equal. Requiring that both gaps do not overlap, enforces a lower bound on the modulation speed. It is shown that although an increase in the modulation index is desirable, as it enhances the non-reciprocal behaviour, it also limits the range of possible modulation speeds. The effective complex refractive index is calculated over a wide frequency range. It is shown that peaks appear in the extinction coefficient, indicating scattering to Stokes' and Anti-Stokes' waves. Finally, a comprehensive numerical analysis based on the Finite Difference Time Domain method is developed to verify and demonstrate the intriguing properties of space-time modulated media.

Bloch-Floquet waves in optical ring resonators

Modal coupling between frequency-degenerate resonances of an optical ring resonator is a commonly observed phenomenon that results in adverse mode splitting. Traditionally, this coupling is attributed to Rayleigh scattering of a propagating electromagnetic wave into its associated degenerate counter-propagating mode from small perturbations to the dielectric material of the resonator. We have chosen to reframe the problem of intracavity Rayleigh scattering by considering the optical ring resonator as an infinitely-long, one-dimensional photonic crystal (PhC) that possesses a lattice constant equal to the perimeter of the ring. Through application of Bloch-Floquet theory, we show that modal coupling between degenerate resonances of a ring can effectively be described as the formation of photonic frequency bands in the dispersion relation of the resonator. We additionally demonstrate that the Bragg planes of the PhC lattice coincide with the phase matching conditions for constructive interference in the ring. Finally, we show that the magnitude of frequency splitting of a particular resonance is proportional to its associated coefficient in the Fourier expansion of the ring's periodic dielectric function.

Field patterns: A new type of wave with infinitely degenerate band structure

Field pattern materials (FP-materials) are space-time composites with PT-symmetry in which the one-dimensional-spatial distribution of the constituents changes in time in such a special manner to give rise to a new type of waves, which we call field pattern waves (FP-waves) (Milton G. W. and Mattei O., Proc. R. Soc. A, 473 (2017) 20160819; Mattei O. and Milton G. W., New J. Phys., 19 (2017) 093022). Specifically, due to the special periodic space-time geometry of these materials, when an instantaneous disturbance propagates through the system, the branching of the characteristic lines at the space-time interfaces between phases does not lead to a chaotic cascade of disturbances but concentrates on an orderly pattern of disturbances: this is the field pattern. In this letter, by applying Bloch-Floquet theory, we show that the dispersion diagrams associated with these FP-materials are infinitely degenerate: associated with each point on the dispersion diagram is an infinite space of Bloch functions. Each generalized function is concentrated on a specific field pattern, each parameterized by a variable that we call the launch parameter. The dynamics separates into independent dynamics on the different field patterns, each with the same dispersion relation.