Bloch-Floquet Theorem Research Articles

Wave propagation in tailored metastructures consisting of elastic beams and rigid bodies.

This paper presents a study of wave propagation through an infinite periodic structure that consists of elastic Timoshenko beams interconnected with rigid bodies. This is a generalized approach in which the beams are not coaxial and the centre of mass of each rigid body is placed away from the intersection of their neutral axes. An analytical approach is used by applying the transfer matrix method (TMM), along with the Floquet-Bloch theorem for elastic wave propagation. Subsequent parametric analysis is performed with visualization of resulting band diagrams of a representative structure. These results are verified through comparison with solutions obtained using the finite-element method (FEM). In this manner, a comprehensive dynamical analysis of tailored metastructures is provided.This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

In plane mechanical properties of hexagonal V-chiral and Tri-chiral metamaterials

The reasonable design of man-made multifunctional metamaterials with advanced mechanical and physical properties that are not found in nature is both challenging and important. In this paper, novel chiral structures are designed by adding V- and Tri- chiral elements to the typical hexagonal honeycomb substrate, so as to realize the modulation of in plane mechanical properties. Finite element simulations show that the Poisson's ratio of the hexagonal V-chiral structure is negative when the strain is greater than 0.4, and that of the hexagonal Tri-chiral structure is always negative. Quasi-static compression tests show that the introduction of chiral properties results in a stronger load-bearing capacity, and the hexagonal Tri-chiral structure has a stronger energy absorption (EA) and specific energy absorption (SEA) capability. A parametric study is carried out by means of FEM, by adjusting the geometrical parameters of the proposed structures, the performance in terms of energy absorption and compressive stiffness can be improved. Subsequently, Bloch-Floquet theorem is employed for the dispersion relations of the chiral metamaterials and the results show that compared to conventional hexagonal honeycomb, chiral metamaterials are able to generate band gaps in a lower frequency range. Numerical simulation results show that the bandwidth of the proposed novel chiral metamaterials can be tuned through a reasonable selection of hexagonal edge length, wall thickness and radius. Experimental investigations are conducted to test the vibration transmission rate of Structure Ⅲ, and the results prove that the structure is capable of vibration suppression in the target frequency band range. Finally, by filling Structure Ⅲ with steel columns, the structure generates band gaps in the lower frequency range, hence the application of this chiral metamaterial is broadened.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

Machine learning-based optimal design of an acoustic black hole metaplate for enhanced bandgap and load-bearing capacity

This paper introduces a novel machine learning-based optimization strategy for multi-functional acoustic black hole (ABH) metaplates. The primary objective is to achieve a multi-functional metaplate with excellent performance in elastic wave attenuation and load-bearing capacity simultaneously. The paper begins by describing the design of nanocomposite ABH metaplates, presenting a new pathway to realize multi-functional metaplates. Then, a semi-analytical method, based on plate theory and the Bloch–Floquet theorem, is introduced to consider the band structure of the nanocomposite metaplates. Through systematic analysis, the impacts of the ABH effect, nanocomposite reinforcements, and the viscoelastic damping layer on the bandgaps and strain energy compliance are highlighted. Meanwhile, two optimization objectives representing bandgap characteristics and in-plane stiffness are derived respectively. Subsequently, a deep learning surrogate model is employed to establish a relationship involving significant parameters with the optimization objectives. The performance evaluation confirms accuracy and computational speed of the surrogate model. Finally, an optimization strategy based on deep reinforcement learning is proposed to obtain multi-functional metaplates with superior bandgaps, enhanced in-plane stiffness, or both. The robustness and efficiency of the strategy are demonstrated under different tests. The results show that the proposed strategy can achieve identical results as the genetic algorithm and nondominated sorting genetic algorithm-II, while surpassing them in computational efficiency and balancing multiple objectives. The findings of this study serve as valuable references for the future development and application of multi-functional advanced metamaterials.

Low-Frequency Bandgap Characterization of a Locally Resonant Pentagonal Phononic Crystal Beam Structure.

This paper proposes a local resonance-type pentagonal phononic crystal beam structure for practical engineering applications to achieve better vibration and noise reduction. The energy band, transmission curve, and displacement field corresponding to the vibration modes of the structure are calculated based on the finite element method and Bloch-Floquet theorem. Furthermore, an analysis is conducted to understand the mechanism behind the generation of bandgaps. The numerical analysis indicates that the pentagonal unit oscillator creates a low-frequency bandgap between 60-70 Hz and 107-130 Hz. Additionally, the pentagonal phononic crystal double-layer beam structure exhibits excellent vibration damping, whereas the single-layer beam has poor vibration damping. The article comparatively analyzes the effects of different parameters on the bandgap range and transmission loss of a pentagonal phononic crystal beam. For instance, increasing the thickness of the lead layer leads to an increase in the width of the bandgap. Similarly, increasing the thickness of the rubber layer, intermediate plate, and total thickness of the phononic crystals results in a bandgap at lower frequencies. By adjusting the parameters, the beam can be optimized for practical engineering purposes.

Coupled bandgaps and wave attenuation in periodic flexoelectric curve nanobeams

Curve beams are widely used in structural engineering like civil engineering and aircraft structures. Due to their coupling effects of bending, shear, and torsion, curve beams have a significant role in wave propagation fields as complete bandgaps can be generated. With the miniaturization of devices, it becomes increasingly imperative to investigate wave characteristics of curve beams at the nanoscale, taking into account the flexoelectric effect. In this study, a theoretical model for the periodic flexoelectric curve beams is established under the strain-gradient electro-elasticity theory. Subsequently, a customized state-space-based transfer matrix method for flexoelectric curve beam is specifically proposed, referred to as FCB-TMM. According to the Floquet-Bloch theorem, the dispersion relations for the periodic flexoelectric curve beams with intriguing coupling characteristics can be obtained and verified. Additionally, the influence of the flexoelectric effect and strain gradient effect on its complete bandgaps is analyzed. The results indicate that the flexoelectric effect plays a vital role in wave propagation at small scales, causing a shift in the bandgaps towards higher frequencies. Ultimately, the synergistic effects of significant geometric and material parameters combined with the flexoelectric effect on the bandgaps are systematically discussed. It is discovered that changing flexoelectric coefficients may alter the pattern of bandgap variation with these geometric and material parameters, while the influence of flexoelectric coefficients on bandgaps varies depending on the values of these parameters. Our investigation offers an insight into understanding the wave propagation of the curve nanobeams, and thus further guides the application and development of flexoelectric wave components in MEMS/NEMS.

Non-reciprocity and selective wave amplification in nonlinear inertia-induced autoparametric periodic structures

Inspired by the spatiotemporally modulated metamaterials, we propose a one-dimensional autoparametric periodic structure induced by nonlinear inertia that is realized simply by constructing an array of pendula coupled to local oscillators. The mechanism of autoparametric modulation is revealed using a unit cell. The evolution equations for a superlattice describing the global dynamics of the wave-vibration coupling are obtained analytically by combining the Floquet-Bloch theorem and the multiple-time scales perturbation technique. The band structure exists corresponding to the fixed point in the phase space, wherein large amplitude local vibrations with distributed traveling phases modulate the inertial properties of the host pendula propagating with a small amplitude signal wave by playing the role of a pump wave. Non-reciprocal dispersion relations are demonstrated both by numerical results and perturbation results with two-order accuracy. Selective wave amplification with two-mode directional emission is observed in a two-port system which is composed of a sandwich composite medium. The present research based on the proposed autoparametric structure paves a way towards building tunable wave manipulators, vibration energy migrators and mechanical amplifiers.

Metamaterial sandwich plates with two-degree of freedom inertial amplified resonators for broadband low-frequency vibration attenuation

Sandwich plates are extensively utilized across various fields, encompassing building engineering, mechanical engineering, and aerospace engineering, owing to their exceptional stiffness-to-weight ratio. However, effectively attenuating the low-frequency and broadband vibrations of these plates poses a significant challenge. This paper proposes a new type of metamaterial sandwich plate that incorporates two-degree of freedom inertial amplified resonators (IA-MSPDF2), to attain two low-frequency band gaps (BGs) and achieve broadband vibration attenuation. The dispersion relation of the IA-MSPDF2 is calculated based on the Bloch-Floquet theorem, and the generation mechanism of two low-frequency BGs is analyzed through eigenmodes. Both numerical and experimental studies are conducted to substantiate the advantages associated with the presence of two BGs in the IA-MSPDF2 design. The results show that the enhanced coupling effect between the primary and secondary resonators of the IA-MSPDF2 leads to the band associated with the local resonance that shifts to lower frequencies, resulting in a Bragg scattering BG that arises above the locally resonant BG. Compared to the metamaterial sandwich plate with one-degree of freedom inertial amplified resonators (IA-MSPDF1) of equal mass, the IA-MSPDF2 exhibits an increased relative bandwidth of BG by 15%. Increasing the damping of the inertial amplified resonator causes two attenuation zones to widen and merge into a wider attenuation zone. The proposed IA-MSPDF2 can robustly and effectively attenuate the low-frequency and broadband vibration with a small mass cost, contributing to the further exploration and utilization of metamaterial sandwich plates in engineering applications.

A metamaterial sandwich plate with spring-lever-mass resonators

Sandwich plates are widely used in practical engineering, including civil engineering, transportation, aerospace, and other industrial fields. However, characteristics of high specific stiffness and strength cause poor performance in the vibro-acoustic fields. Therefore, resonators are introduced into the core of sandwich plates to improve their performance on vibration isolation. This paper proposes a new metamaterial sandwich plate, multiple spring-lever-mass resonators are introduced into the sandwich plate to obtain the low-frequency band gaps. The band structure of the metamaterial sandwich plate with spring-lever-mass resonators is calculated by the finite element method with the Bloch-Floquet theorem, and the band gap of the unit cell is obtained. The transmission spectra of the proposed structure are investigated to verify the accuracy of prediction results. The results show that effective performance on low-frequency vibration isolation via the metamaterial sandwich plate with spring-lever-mass resonators can be achieved, and this study can provide valuable references for related research and potential applications on the manipulation of wave propagation.

Projective spacetime symmetry of spacetime crystals

Wigner’s seminal work on the Poincaré group revealed one of the fundamental principles of quantum theory: symmetry groups are projectively represented. The condensed-matter counterparts of the Poincaré group could be the spacetime groups of periodically driven crystals or spacetime crystals featuring spacetime periodicity. In this study, we establish the general theory of projective spacetime symmetry algebras of spacetime crystals and reveal their intrinsic connections to gauge structures. As important applications, we exhaustively classify (1,1)D projective symmetry algebras and systematically construct spacetime lattice models for them all. Additionally, we present three consequences of projective spacetime symmetry that surpass ordinary theory: the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless Floquet crystals, and symmetry-enforced crossings in the Hamiltonian spectral flows. Our work provides both theoretical and experimental foundations to explore novel physics protected by projective spacetime symmetry of spacetime crystals.

Wave propagation analysis for three typical box girders: Dominant wave modes and their applications

Because of the prevalent application of bridges in transportation, problems such as vibration and noise of bridges, especially in densely populated urban areas, are increasingly arousing concerns. The vibro-acoustic behaviors of bridges are important for vibration and noise control. Although these vibro-acoustic behaviors have been reported and discussed in several studies in terms of modal analysis, this study focuses on the probe into the vibration properties of three typical box girders (BGs) from the wave motion perspective: concrete, steel–concrete composite, and steel BGs. The dispersion characteristics and mode shapes of waves propagating in BGs are identified using the wave finite element method (WFEM). This calculation method combines the finite element method (FEM) with the Bloch–Floquet theorem, and it is validated by the existing literature and FEM. The wave propagation characteristics of the three typical BGs are then compared to analyze their similarities and differences in vibration. Furthermore, the contribution of each wave mode to the whole vibration and the local vibration of BGs is determined; consequently, the important vibration mechanism can be identified and ranked in different frequency bands. Lastly, some feasible application of vibration control for BGs is given in the light of the vibration mechanism. The study results provide theoretical and practicable foundation of vibration control for BG bridges in urban rail trafficsystems

Attenuation of Rayleigh and pseudo surface waves in saturated soil by seismic metamaterials

Seismic metamaterials have received extensive research interest due to their bandgap properties, simplicity in design principles, and stability in response. They have been developed to protect buildings or architectures susceptible to damage from surface elastic waves. In practice, the ground soil is generally a multiphase medium, and the influence of its permeability and viscosity on seismic metamaterials is not yet clear. In this work, we developed a formulation that combines Biot’s theory and Bloch-Floquet theorem to investigate the complex band structures and transmission properties of Rayleigh and pseudo surface waves (PSWs) for pillared and inclusion-embedded seismic metamaterials in saturated soil. It is shown that the ratio of fluid viscosity and permeability η/κ have an impact on the surface wave attenuation and the performances of seismic metamaterials, where the smaller ratio benefits the surface wave broadband attenuation and metamaterials attenuating effects. The complex band structures reveal that inclusion-embedded metamaterials can support the propagation of PSWs having a phase velocity higher than that of the transverse bulk waves. The PSWs are significantly affected by the rubber viscosity due to the mode displacements concentrated in the rubber coatings. The higher viscosity of metamaterials also allows for broadband attenuation of Rayleigh surface waves. The results of this study will present an appropriate way to design viscoelastic seismic metamaterials in saturated soil for low-frequency surface wave attenuation.

Anisotropy and dispersion of elastic waves in periodically multilayered anisotropic media

The propagation characteristics of elastic waves in general periodically multilayered media formed by alternate arrangement of two or more arbitrarily-anisotropic materials are comprehensively studied. Combining with the state-space formalism for obtaining the wave solutions to the constituent layers, the Floquet-Bloch theorem for providing the periodic relations between neighboring unit cells, the method of reverberation-ray matrix (MRRM) is employed to derive the general dispersion equations of the periodically multilayered anisotropic media. By solving the dispersion equations in cases of all possible sorts of wavenumber components, the complete anisotropic and dispersion characteristics of elastic waves can be provided. By numerical examples, the slowness and the phase velocity curves are provided to summarize the innovative anisotropic characteristics of elastic waves in three coordinate planes. The effects of the anisotropic parameters on the slowness/phase velocity curves have also been discussed. Moreover, all kinds of dispersion curves including the frequency-related and phase velocity-related curves along the directions perpendicular, parallel, and oblique to the layering are provided to reflect the general dispersion characteristics of elastic waves from various viewpoints. It is found that the innovative anisotropy characteristics of elastic waves in the periodically multilayered anisotropic media include that the slowness curves have periodicity in the thickness direction and the least positive period decreases with the increasing of frequency. As the frequency becomes big enough, the slowness/phase velocity curves show interactions between the modes within the overlapping neighboring periods. The common dispersion characteristics of elastic waves along various directions include the symmetry of wavenumber, the asymptotic of the wavelength and the phase velocity curves to the frequency lines corresponding to zero wavenumber, as well as the cutoff property of the phase velocities of the lowest three basis modes. The waves in oblique directions are special in that the mode transition and repulsion due to the zone folding effect of periodicity along thickness as well as the cutoff property of higher-than-third-order overtones along layering are simultaneously in effective.

Analysis of bending waves and parametric influence on band gaps in periodic track structure

Vibration of railway tracks generated due to rail-wheel interaction is a key research concern. A rail with equally spaced supports can be regarded as a periodic structure. Such a structure possesses unique property of band gap because of that they act as vibro-acoustic filters. Here, the propagation behavior of vertical and lateral bending waves is studied in a periodic ballastless track structure. Bloch-Floquet theorem is used to formulate the dispersion relations for the propagation of two types of waves which is validated by corresponding finite element model. The results show coexistence of both resonance and Bragg type band gaps they are owing to locally resonant units and spatial periodicity in the track. Furthermore, the study also explores the impact of track parameters, such as pad stiffness and sleeper spacing, on the properties of these band gaps.

Effects of corrugated boundaries on Rayleigh waves in a piezoelectric semiconductor substrate covered with a metal layer

In the present work, analytical solutions for Rayleigh waves propagating in a metal covered layer over a piezoelectric semiconductor (PSC) substrate with corrugated surface and corrugated interface are obtained. The solutions of the various physical fields can be expressed as a summation of infinite harmonic waves by using the Bloch–Floquet theorem due to the existence of the periodic corrugated surface and corrugated interface. The effects of the height and contour shape of the corrugated surface and corrugated interface, the doping density of the PSC substrate and the thickness of the metal covered layer on the dispersion and attenuation curves and mode shapes of Rayleigh waves are discussed via numerical example. The results show that the existence of the corrugated surface will lead to the decrease of the phase velocity and non-monotonous change with wavenumber, besides, and significantly increases the attenuation near k0=π/Λ(k0 and Λ are the wavenumber and period, respectively). In addition, the effects of periodic corrugated surface are much greater than those of periodic corrugated interface and are closely related with the ratio of the wavelength to the structure period. Moreover, the sinusoidal corrugated surface can be used to better control the propagation characteristics of Rayleigh waves by comparison.

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

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

Shear horizontal wave propagation on a piezoelectric semiconductor substrate under slight ridge or thin metal strip gratings

In the present work, analytical solutions for shear horizontal (SH) waves propagating in an n-type piezoelectric semiconductor (PSC) substrate under periodic slight ridge or thin metal strip gratings are obtained. The solutions of the various physical fields can be expressed as a summation of infinite harmonic waves by using the Bloch–Floquet theorem due to the existence of the periodic ridge or metal grating. Based on the numerical calculations, the effects of the width, height and shape of the ridge, the doping density of the PSC substrate and the material properties of the metal grating on the dispersion and attenuation curves of the SH waves are investigated. The results show that both the periodic ridge and metal grating have evident effects on the propagation characteristics and displacement mode shapes of the SH waves and the effects are closely related with the ratio of the wavelength to the structure period. In addition, the metal gratings can better control the propagation characteristics of SH waves than the ridges, namely, the effects of the metal gratings are more obvious and flexible than that of the ridges.

Quasi-static band gaps in metamaterial pipes with negative stiffness resonators

A novel configuration of periodically supported pipes with negative stiffness resonators (NSRs) at midspans is investigated in this paper. By leveraging the expanded bandwidth and enhanced damping effects offered by the NSRs, as well as the quasi-static (QS) band gaps of periodically supported pipes, this configuration enables the realization of ultrawide QS band gaps. The Floquet-Bloch theorem and Finite Element Method (FEM) are employed to determine the dispersion relationship of the infinite periodic pipe, elucidating the wave propagation characteristics. The band formation mechanism is revealed by conducting a comparative analysis with other systems in which unit cells only comprise periodic supports or NSRs. By analyzing the dispersion relationships, the effects of critical parameters on the transition and interplay between Bragg and local resonance (LR) band gaps are investigated. The results reveal that, with increasing absolute values of the negative stiffness ratio or mass ratio at a constant resonant frequency, the LR band gap created by NSRs eventually transforms into a wide QS band gap. The study further extends to metamaterial pipes of finite length, where FEM simulations are carried out using ANSYS to confirm the physical band-gap properties observed in the periodic systems. The advantage of the wave attenuation bandwidth introduced by NSRs is demonstrated by comparing the results with those obtained by traditional types of resonators. With the same added mass, the system with NSRs exhibits a substantially wider LR band gap than the traditional resonators. Finally, owing to the damping magnification mechanism of NS elements, when a reasonable amount of damping is introduced, the proposed system shows superiority in achieving ultrawide QS band gaps through band-gap merging over pipe systems with suspended resonators.

Orthogonal analysis of space-time crystals.

This paper presents a space-time-wise orthogonal analysis of space-time crystals. This analysis provides a solution consisting of a pair of explicit parametric equations that result from a separate application of the Bloch-Floquet theorem in the (orthogonal) directions of space and time. Compared with previous approaches, this solution offers the benefits of greater simplicity, clearer emphasis on space-time duality, and deeper physical insight.