Bloch Theory Research Articles

Rest length controlled tunable band structure in periodic tensegrity metastructure – a numerical study using a novel consistent stiffness and mass formulation

Elastic wave propagation in various periodic metastructures and its application to wave filtering, vibration isolation, acoustic cloaking, etc., have gained significant attention among researchers in recent years. Despite the presence of band gaps, traditional lattice structures lack dynamic tunability. The solution lies in using tensegrity as unit cells. This article presents analysis of tunable wave propagation through a metastructure formed by tessellation of a 2D tensegrity unit cell in one dimension. Equations of motion of the unit cell are derived with consideration of flexibility and distributed mass of the bars, which is often neglected in the literature. The formulation is extended for wave dispersion analysis using the Floquet–Bloch theory. Next, the order of the formulation is reduced for decoupled analysis of symmetric and antisymmetric modes. The dispersion relations for different modes have been validated with nonlinear as well as linearized frequency response analysis under impulse load. Next, the effect of rest length of strings and alternating arrangement of bars on band gaps is studied in detail.

Exact solution to band structures of one-dimensional viscoelastic functionally graded phononic crystals

In this article, longitudinal wave (P-wave) band structures of functionally graded viscoelastic phononic crystals (PnCs) with exponential distribution properties are analytically investigated via an exact method. Differing from the laminated model, displacement and stress vectors of normal incident P-wave are obtained exactly by utilizing a stress-transform technique and the state space method. According to the continuous condition and Bloch theory, the eigenvalue equation of band structure is derived by applying transfer matrix method. The storage modulus is considered to be frequency-dependent while the loss modulus is neglected for the consideration of components viscoelasticity. Dispersion curves, obtained from the calculation of eigenvalue equation, are validated by the results of exist laminated model and transmission spectra which are also present to reveal the propagation characteristic of normal incident P-wave. Numerical examples indicate that band width and center frequencies of band gaps can be effectively altered by changing magnitude of exponential function, ratio between initial modulus and final modulus. Other new band gaps are opened as well as wider band width are achieved with the increase of gradient index к . The exact analytical solution presented in this paper can be the benchmark for those approximation solutions.

Long-wave instabilities of a power-law fluid flowing over a heated, uneven and porous incline: A two-sided model

The stability conditions of a two-dimensional gravity-driven flow of a thin layer of a power-law fluid flowing over a heated, uneven, inclined porous surface are investigated. A two-sided model is employed to account for the bottom filtration in the porous layer. The governing equations are reduced under the long-wave approximation and the cross-stream dependence is eliminated by means of the Integral Boundary Layer technique. Floquet–Bloch theory is used to investigate at linear level how the porous bottom waviness influences the thermocapillarity stability of the flow in a shear-thinning fluid. Differently from the even case, the linear stability analysis suggests that for flow over sufficiently wavy undulations the thermocapillarity may stabilize the equilibrium flow, depending on the values of dimensionless governing numbers and parameters. This stabilizing phenomenon is enhanced by the shear-thinning rheology of the fluid while it is reduced by the permeability of the layer. Numerical simulations, performed solving the reduced nonlinear model through a second order Finite Volume scheme, confirm the results of the linear stability analysis.

Graphene-Tuned, Tightly Coupled Hybrid Plasmonic Meta-Atoms.

Tightly coupled meta-atoms (TCMAs) are densely packed metamaterials with unnatural refractive indexes. Actively modulated TCMAs with tunable optical properties have found many applications in beam shaping, holography, and enhanced light-matter interactions. Typically, TCMAs are studied in the classic Bloch theory. Here, tightly coupled H-shaped meta-atoms are proposed with an ultra-high permittivity of ~6000, and their active modulation with graphene is designed by using the tightly coupled dipole array (TCDA) theory. The H-shaped meta-atoms are used as dipole arms, and the graphene strips function as the dipole loads. By tuning the chemical potential of graphene, the resonant amplitude, frequency, and permittivity are dynamically modulated. The simulations indicate that the real and imaginary parts of permittivity change from 6854 to 1522 and from 7356 to 2870, respectively. The experimental validation demonstrates a modulation depth of 11.6% in the resonant frequency, i.e., from 219.4 to 195 GHz, and a substantial 52.5% modulation depth in transmittance under a bias voltage of less than 1.5 V.

Band gap characteristics of bionic acoustic metamaterials based on spider web

For the purpose of low-frequency vibration isolation and noise reduction, combined with the nested mode of spider web, a new concave star bionic acoustic metamaterial was proposed in this paper. Based on the finite element method (FEM) and Bloch theory, the effects of vertex connection mode, nesting level (n), and rotation angle (α) on the band gap distribution, transmission spectrum, and isofrequency line of concave star structure were discussed. The results showed that the two-dimensional (2D) concave star structure had the lowest opening frequency of the first band gap when the vertices coincide and n = 3; when α = 45°, the three-dimensional (3D) concave star structure had the widest band gap distribution and the lowest first band gap in the frequency range within 1000 Hz. The correctness of the simulation results was verified by experiments. These findings can provide a new idea for low-frequency vibration isolation and noise reduction.

Stability of a Multi-Span Current-Carrying Beam Resting on Periodic Supports and Exposed to an External Magnetic Field

Abstract The stability of a current-carrying beam in an external magnetic field is studied in this paper. The beam rests on periodic supports and is modeled as an Euler–Bernoulli elastic beam. Based on the linearized stability equation and propagator transfer matrix approach for a multi-span finite-length beam, the stability equation and solutions of the several boundary value problems are obtained. While the Floquet–Bloch theory is widely used in the dynamic problem of phononic and photonic structure conditions, its application to the solution of infinite length beam static stability problem is novel. The stability values of the periodic infinite beam model are in very good agreement with one of the finite-length multi-span beams.

Uncovering exceptional contours in non-Hermitian hyperbolic lattices

Hyperbolic lattices are starting to be explored in search of novel phases of matter. At the same time, non-Hermitian physics has come to the forefront in photonic, optical, phononic, and condensed matter systems. In this work, we introduce non-Hermitian hyperbolic lattices and elucidate its exceptional properties in depth. We use hyperbolic Bloch theory to investigate band structures of hyperbolic lattices in the presence of non-Hermitian on-site gain and loss as well as non-reciprocal hopping. Using various analytical and numerical approaches we demonstrate widely accessible and tunable exceptional points (EPs) and contours in {10,5} tessellations, which we characterize using phase rigidity, energy scaling, and vorticity. We further demonstrate the occurrence of higher-order EPs and contours in the {8,4} tessellations using the method of Newton polygons, supported by vorticity and phase rigidity computations. Finally, we investigate the open boundary spectra and densities of states to compare with results from band theory, along with a demonstration of boundary localisation. Our results unveil an abundance of exceptional degeneracies in hyperbolic non-Hermitian matter.

A Delicate Balance Between Spin-Wave Mediated Weak Localization and Electron-Phonon Scattering in the Design of Zero Temperature Coefficient of Resistivity.

Zero thermal coefficients of resistivity (ZTCR) materials exhibit minimal changes in resistance with temperature variations, making them essential in modern advanced technologies. The current ZTCR materials, which are based on the resistivity saturation effect of heavy metals, tend to function at elevated temperatures because the mean free path approaches the lower limit of the semiclassical Boltzmann theory when the temperature is sufficiently high. ZTCR materials working at low-temperatures are difficult to achieve due to electron-phonon scattering, which results in increased resistivity according to Bloch's theory. In this work, the ZTCR behavior at low-temperatures is realized in pre-microstrained Mn3NiN. The delicate balance between the resistivity contribution from electron-phonon scattering and spin-wave mediated weak localization is well revealed. A remarkable temperature coefficient of resistivity (TCR) value as low as 1.9ppmK-1 (50K≤T≤200K) is obtained, which is significantly superior to the threshold value of ZTCR behavior and the application standard of commercial ZTCR materials. The demonstration provides a unique paradigm in the design of ZTCR materials through the contraction effects of two opposite conductance mechanisms with positive and negative thermal coefficients of resistivity.

Modal analysis of waveguide for the study of frequency bandgaps of a bounded periodic medium

A novel method based on waveguide modal analysis is presented to evaluate the effectiveness in terms of frequency bandgaps of bounded periodic metamaterials. The effect of the size of the bounded medium, i.e., the number of periodic unit cells, on the frequency bandgaps predicted by Floquet–Bloch theory in the infinite case is investigated and quantified. The waveguides considered in this work have a bounded cross-section and an infinite longitudinal axis simulated by Perfectly Matched Layers. Its useful area is made of a bounded 1D or 2D-periodic matrix-inclusion metamaterial. In the 2D-periodic case, an approach combining the waveguide modal analysis with the Floquet–Bloch transform is further proposed. Applying the proposed method, the SH-waves propagating in a waveguide are studied by using the finite element method. The discrete spectrum of the waveguide and the associated attenuated or trapped wave modes are calculated and analyzed, which provides, within a semi-analytic framework, an exact characterization regarding the attenuation coefficient and the frequency bandgaps of the studied bounded periodic medium. More particularly, effectiveness indicators are defined to compare the width and the position of frequency bandgaps between the bounded and infinite cases as a function of the cell number. Last but not least, the proposed method is also successfully applied to bounded periodic media with local resonators to characterize their filtering and attenuation effectiveness, which shows its interest in such a case of great practical interest.

Homogenization of Schrödinger equations. Extended effective mass theorems for non-crystalline matter

This paper concerns the homogenization of Schrödinger equations for non-crystalline matter, that is to say the coefficients are given by the composition of stationary functions with stochastic deformations. Two rigorous results of so-called effective mass theorems in solid state physics are obtained: a general abstract result (beyond the classical stationary ergodic setting), and one for quasi-perfect materials (i.e. the disorder in the non-crystalline matter is limited). The former relies on the double-scale limits and the wave function is spanned on the Bloch basis. Therefore, we have extended the Bloch Theory which was restricted until now to crystals (periodic setting). The second result relies on the Perturbation Theory and a special case of stochastic deformations, namely stochastic perturbation of the identity.

Applicability and limitations of cluster perturbation theory for Hubbard models

We present important use cases and limitations when considering results obtained from cluster perturbation theory (CPT). CPT combines the solutions of small individual clusters of an infinite lattice system with the Bloch theory of conventional band theory to provide an approximation for the Green’s function in the thermodynamic limit. To this end, we are investigating single-band and multi-band Hubbard models in 1D and 2D systems. A special interest is taken in the supposed pseudogap regime of the 2D square lattice at half-filling and intermediate interaction strength (U≤3t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U \\le 3t$$\\end{document}) as well as the metal–insulator transition. We point out that the finite-size level spacing of the cluster limits the resolution of spectral features within CPT. This restricts the investigation of asymptotic properties of the metal–insulator transition, as it would require much larger cluster sizes that are beyond computational capabilities.

A high-order TE50 mode waveguide with single mode transmission property and its application in slot array antenna

Conventional array antenna techniques contain various feeding networks, such as T-junction power dividers, phase dividers, and baluns. To feed a relatively large array, one can cascade the traditional T-junction power dividers, resulting in the increase of design complexity. In this paper, an overmoded waveguide supporting the single mode propagation of TE50 mode, is presented. Since TE50 mode can be separated into five TE10 modes with equi-amplitude and opposite phase properties, it is advisable to distribute the input signal flexibly and decrease the cascading of the T-junctions for the traditional feeding networks. To actualize a TE50 mode waveguide, several rows of metallic blocks acting as metamaterial are periodically loaded inside an overmoded waveguide to tailor the dispersion of the waveguide modes and yield the proposed mode bandgaps. Considering the Bloch theory, parts of the undesired waveguide modes are not supported inside the metamaterial and will be removed by these bandgaps. In addition, central feeding method is used to suppress the unwanted modes further in light of mode mismatch. As a proof-of-concept, we drilled the slot array on the top of the overmoded waveguide directly and give rise to a pencil beam radiation along the broadside with 15–17 dBi realized gain among 9.4–10.6 GHz, which demonstrates the stable feeding ability of TE50 mode.

A semi-analytical method for characterization of fractal spoof surface plasmon polaritons with a transfer matrix and bloch theory

In this paper, a semi-analytical approach is introduced to analyze a spoof plasmonic structure, with an arbitrary geometry. This approach is based on a combination of techniques that employ a full-wave simulator and the Bloch theorem. By applying periodic boundary conditions, the real and imaginary parts of the equation obtained from the equivalent network have been calculated. To show the accuracy and validity of this proposed approach, a complementary Minkowski fractal SSPP unit cell has been designed and analyzed, and this has been used in a surface plasmonic transmission line. The results of our proposed method have been compared to measured results, and the simulated and measured results showed that the SSPP transmission line possesses high performance, from 1.45 to 5 GHz.

Exact analytical solution for shear horizontal wave propagation through locally periodic structures realized by viscoelastic functionally graded materials

The paper presents a novel comprehensive exact analytical solution for modeling linear shear-horizontal (SH) wave propagation in an isotropic inhomogeneous layer made of functionally graded material, using local Heun functions. The layer is a composite of two materials with varying properties represented by spatial variations following the square of the sine function. The Voigt–Kelvin model is used to account for material losses. The study focuses on SH waves incident at a specific angle and employs the wave splitting technique to analyze forward and backward waves, facilitating the computation of reflection and transmission coefficients at any point in the inhomogeneous structure. The proposed solution utilizes the periodic nature of material functions and employs the Floquet–Bloch theory to derive an exact analytical solution. This approach is particularly suited for cases where SH waves encounter locally periodic functionally graded material. A Riccati equation-based verification is conducted to compare the frequency-dependent modulus of the reflection coefficient obtained from the analytical solution with numerically solved results. The presented work provides a comprehensive and versatile analytical solution for studying linear SH wave propagation in locally inhomogeneous isotropic layers, contributing to the theoretical understanding of elastic wave fields and practical applications.

High frequency multi-field continualization scheme for layered magneto-electro-elastic materials

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.

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Acoustical characteristics of solid elastic mediums with oriented microfructures

Model for interaction of plane elastic longitudinal waves with contains oriented microfractures with specific boundary conditions for “linear slip@ approximation is developed. The form of dispersion equation is defined in approximation of Bloch and Floque waves theory. Results of its solution which define character of functions for propagation phase velocity and attenuation coefficient of effective longitudinal wave in dependence on crack faces microfracture parameters and other model parameters.

Shape and non-shape unified isogeometric optimal design for extremal band gaps

A unified shape and non-shape design optimization for lattice structures is performed to obtain extremal band gaps in an isogeometric computational framework, which uses the same NURBS basis functions as used in CAD description. Using a precise numerical model in both response and shape sensitivity analyses, enhanced response and sensitivity can be obtained, which leads to a precise optimal design. A plane wave propagation in infinite periodic lattice is analyzed within a representative unit cell using the Bloch theory with periodic boundary conditions. The ligaments of lattice structures are modeled using geometrically exact beams whose cross-sectional area and configuration are regarded as design variables, which are parameterized by the NURBS basis functions. The design variables are controlled by a mathematical programming algorithm with efficient adjoint design sensitivity. The obtained optimal lattice structures of triangular, squared, hexagonal, and Kagomé structures are fabricated using a 3D printing machine to be validated by physical experiments. Through harmonic response analysis and physical experiment for the optimal and the original designs, the wave attenuation properties are compared to demonstrate the validity of the obtained optimal design.

Band Gap Analysis for Materials with Cookie-Shaped Auxetic Microstructures, Using Finite Elements

Microstructures, including auxetic ones, influence the overall behavior of a material. Particularly during wave propagation, band gaps may appear. This study focuses on the discovery and analysis of the band gaps that occur in a material with auxetic microstructures of the cookie type, which have certain advantages with respect to production and fatigue resistance. Finite element analysis at a representative volume element, using Bloch theory, and verification with a plate made of microstructures at various assemblages has been used.

Nonlinear exceptional-point lasing with ab initio Maxwell–Bloch theory

We present a general analysis for finding and characterizing nonlinear exceptional point (EP) lasers above threshold using steady-state ab initio Maxwell–Bloch equations. For a system of coupled slabs, we show that a nonlinear EP is obtained for a given ratio between the external pumps in each resonator and that it is associated with a kink in the output power and lasing frequency, confirming coupled-mode theory predictions. Through numerical linear stability analysis, we confirm that the EP laser can be stable for a large enough inversion relaxation rate. We further show that the EP laser can be characterized by scattering a weak signal off the lasing cavity so that the scattering frequency spectrum exhibits a quartic divergence. Our approach can be applied to arbitrary scatterers with multi-level gain media.