Bloch Wave Analysis Research Articles

Rainbow trapping of out-of-plane mechanical waves in spatially variant beam lattices

We numerically and experimentally investigate the propagation of mechanical waves in two-dimensional periodic and spatially graded elastic beam lattices. Experiments on metallic lattices admit the characterization of the linear elastic wave dispersion over a wide range of frequencies, resulting in complete, experimentally-constructed dispersion surfaces in excellent agreement with predictions obtained from finite element-based Bloch wave analysis. While Timoshenko beam theory is shown to be sufficiently accurate for predicting the lowest modes, experiments prove that solid finite elements are required to capture the dispersion relations at higher frequencies as well as when mode coupling occurs. Based on an improved numerical procedure, group velocity maps further highlight the directionality of wave dispersion and allow for the simple identification of bandgaps. In addition to classically studied periodic trusses, we extend the framework to spatially graded structures and demonstrate acoustic rainbow trapping in beam lattices undergoing out-of-plane vibrations. Our experiments confirm broadband vibration attenuation of the typical meta-wedge type previously observed only in optics and few mechanical studies. Results further show convincing agreement between Bloch theory-based predictions, finite element simulations, and experimental measurements. Such spatially-variant architected lattices show great promise for steering the motion of elastic waves in applications from wave guiding and wave shielding to energy harvesting.

Dynamics of elastic lattices with sliding constraints

This study investigates the impact of sliders – constraints acting on elastic rods allowing for a transverse displacement jump while maintaining axial and rotational displacement continuity – on the dynamics of a periodic elastic grid, including the effects of axial preload. The grid is linearly elastic and subject to in-plane incremental deformation, involving normal and shear forces and bending moment. The periodicity of the infinite grid permits a Floquet–Bloch wave analysis and a rigorous dynamic homogenization, leading to an equivalent prestressed elastic solid. The investigation is complemented by ad hoc developed F.E. simulations and perturbations with a pulsating Green’s function. Results show that the sliders create band gaps, flat bands and Dirac cones in the dispersion diagrams and generate macro-instability even for tensile prestress. The latter corresponds to the loss of ellipticity at the parabolic boundary in the equivalent elastic solid and provides a rare example of an almost unexplored form of material instability. Therefore, our results offer design strategies for metamaterials and architected materials showing reversible material instabilities and filtering properties for mechanical signals.

Homogenized elasticity and domain of linear elasticity of 2D architectured materials

This article presents a generic method to determine analytically the equivalent homogeneous linear elastic behavior of any 2D architectured material and to determine its macroscopic linear elasticity domain. The method is applied throughout the paper in the equilateral triangular case, as an example. Additional examples of square, isosceles right-angled triangular and hexagonal architectured materials are presented as a conclusion. The proposed method combines a periodic homogenization for the equivalent homogeneous linear elastic behavior and three criteria to establish the macroscopic linear elasticity domain. The first criterion corresponds to the limit strength of the bars. The second one, which corresponds to the apparition of a periodic buckling, involving a finite number of cells, is studied on a single primitive unit cell using Bloch wave analysis. The buckling modes that appear under equibiaxial compression for the equilateral triangle architectured material are similar to experimental results reported in the literature. The third criterion is the aperiodic buckling, or macroscopic instability, which corresponds to the apparition of shear bands or a macroscopic buckling. The resultant macroscopic linear elasticity domain resulting from these three criteria is presented, for the first time, in the case of the equilateral triangle, isosceles right-angled triangle architectured material and also for two other geometries, square and hexagonal architectured material. It is shown that the theoretical properties of these architectured materials are, in an Ashby chart, in the domain of light but relatively strong materials, a domain in which man-made materials are currently poorly represented.

Thermocapillary Patterning of Highly Uniform Microarrays by Resonant Wavelength Excitation

For decades, researchers have been exploring pattern-forming instabilities in free surface thin films in the hope of developing alternative lithographic techniques for applications only requiring resolution limits in the submicron range. Previous studies have shown how the pitch and shape of elements in an array can be varied by adjusting the magnitude of surface forces and growth time prior to solidification in situ. Since the formations emerge naturally from an initial flat molten film, the final arrays exhibit ultrasmooth interfaces and are therefore ideally suited to beam-shaping applications such as thin-film micro-optics. Progress in this field has stalled, however, due to the very nature of the formation process. Even when great care is taken to ensure that initial films are defect-free, final arrays still exhibit unacceptable variability in pitch, shape, and height due to ubiquitous sources of noise responsible for instability and growth. In this work, we focus on a thermocapillary instability in slender molten films exposed to a very large thermal gradient. We begin with a discussion and demonstration of why this instability inextricably leads to highly disordered arrays even if initialized by a film with very small amplitude surface roughness. We then demonstrate how spatially periodic modulation of the thermal field, implemented in three different ways, can induce synchronous growth of highly uniform periodic arrays despite noisy initial conditions. Results based on linear and weakly nonlinear stability analysis, Bloch wave analysis, and direct numerical simulation of the interface equation reveal how resonant wavelength excitations occurring between the modulation and instability driving fields are responsible for such rapid and coherent growth. An additional benefit is that the modulation field can be selected to yield an array pitch much smaller than in unmodulated systems.

Homogenization of an acoustic coating with a steel backing subject to an obliquely incident sound.

An effective homogenization model for the acoustic coating of underwater structures is important for reducing the complexity of acoustic scattering computation, which arises from the huge difference in scale between the integral structure and the inhomogeneous microstructure of the coating. The main difficulty of this homogenization arises from the oblique-incidence effect of external sound waves and the interface effect between the coating and backing. In this work, a hybrid method, combining the Bloch wave analysis and retrieval technique, is proposed to characterize the acoustic behavior of the voided coating backed with a steel plate under the action of external sound waves with an arbitrary incident angle. The effectiveness of this method is validated by numerical simulations and comparison with the Bloch wave method and the traditional retrieval method. The influence of the shear-wave effect under obliquely incident sound waves and the coupling effect between the coating and the backing on the homogenization model is investigated in detail, providing a comprehensive understanding of the effective acoustic behavior of the coating.

Calculating bandgaps of nonlinear mechanical metamaterials

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

Negative dispersion of a form birefringence in subwavelength gratings.

An achromatic response is required in most optical systems for wideband and straightforward configurations. The chromatic response of the optical system depends on the optical dispersion of the elements in the system. Here we study the dispersion of subwavelength grating (SWG) known to have a form birefringence. The birefringence of SWG was numerically analyzed with Bloch wave analysis (BWA) and finite element method (FEM). The sandwiched SWG with two identical substrates was studied for practical applications. We successfully demonstrated the negative dispersion form birefringence of SWG with an optimal duty cycle. This extraordinary dispersion was also shown considering the intrinsic dispersion of materials. Dispersion- and the angular response were in a tradeoff relationship while they depended on periodicity. The optical interference between the grating and the substrates can be eliminated by controlling the duty cycle. Our analysis offers optimal SWG with achromatic birefringence and high transparency, promising in the widespread applications of polarization control devices.

Multi-stable acoustic metamaterials with re-configurable mass distribution

This article proposes a design strategy for acoustic metamaterial lattices which leverages the characteristic multi-stability and kinematic amplification of the internal architecture to realize a re-configurable effective mass distribution, enabling a corresponding frequency band structure of extreme tunability. The approach builds metamaterial lattices from bi-stable elastic elements featuring the typical two-bar inertial amplification mechanism, whose deformation axis is uniquely misaligned with that of its elastic support such that the realized amplification is configuration-specific. Conveniently, a change in the configuration does not simultaneously alter the element dimensions and, therefore, the size/shape of the finite metamaterial structure remains constant. Moreover, as the multi-stability and kinematic amplification are each based on geometry, scaling the resulting metamaterial structure is expected to be straightforward. The dynamic performance of 1D/2D metamaterial architectures exhibiting the proposed design strategy is analytically determined via Bloch wave analysis and supported by numerical demonstration of the corresponding finite structures.

Band gap tuning through microscopic instabilities of compressively loaded lightened nacre-like composite metamaterials

Large deformation effects give the opportunity for tuning and manipulating the dynamic properties of periodic microstructured materials, including those related to vibration control and band gap phenomena. We investigate the wave propagation properties in bioinspired nacre-like composite metamaterials subjected to uniaxial loading processes, also considering the onset of instabilities at the microscopic and macroscopic scales. The dynamic response of a new 2D lightened brick-and-mortar microstructure containing periodically arranged cavities is investigated, by analyzing the propagation of elastic waves superimposed on a deformed state by means of a Bloch-wave analysis. The band gap properties of the proposed advanced metamaterial were investigated in terms of spectral quantities as a function of the main geometrical parameters and of the level of applied deformations. Numerical results have shown that the elastic wave propagation is strongly influenced by the prestress state inducing microscopic instability, thus providing new opportunities to design periodic bioinspired composite metamaterials characterized by wave absorption capabilities as a function of microstructural evolution.

Wave propagation in a continuum nonlinear phononic material with soft nonlinear elastic layers

This presentation discusses modeling the propagation of nonlinear waves in a continuum phononic material. This system is shown to exhibit amplitude-dependent response and energy transfer between frequencies, which are well studied phenomena of nonlinear discrete systems, but are relatively unexplored in fully elastic continuum models. The phononic material is composed of repeated layers of stiff and soft materials, and the nonlinearity is introduced through a nonlinear hyperelastic Gent model in the soft material. We analyze the dispersion of the linearized system using Bloch-wave analysis to identify band gaps in the system, and we use full-scale time-domainfinite element simulations to analyze the dynamic behavior of the system. Generation of zero frequency and second harmonic frequency amplitudes as well as their accumulation with distance are observed, and the influence of phononic band gaps on nonlinear wave propagation are described. We explore the influence of material nonlinearity in soft materials in continuum phononic materials as a mechanism for control of nonlinear wave propagation.

A computational framework for homogenization and multiscale stability analyses of nonlinear periodic materials

AbstractThis article presents a consistent computational framework for multiscale first‐order finite strain homogenization and stability analyses of rate‐independent solids with periodic microstructures. The homogenization formulation is built on a priori discretized microstructure, and algorithms for computing the matrix representations of the homogenized stresses and tangent moduli are consistently derived. The homogenization results lose their validity at the onset of first bifurcation, which can be computed from multiscale stability analysis. The multiscale instabilities include: (a) microscale structural instability calculated by Bloch wave analysis; and (b) macroscale material instability calculated by rank‐1 convexity checks on the homogenized tangent moduli. Implementation details of the Bloch wave analysis are provided, including the selection of wave vector space and the retrieval of real‐valued buckling mode from complex‐valued Bloch wave. Three methods are detailed for solving the resulted constrained eigenvalue problem—two condensation methods and a null‐space projection method. Both the homogenization and stability analyses are verified using numerical examples including hyperelastic and elastoplastic periodic materials. Various microscale buckling phenomena are demonstrated. Aligned with theoretical results, the numerical results show that the microscopic long wavelength buckling can be equivalently detected by the loss of rank‐1 convexity of the homogenized tangent moduli.

An FFT-based approach for Bloch wave analysis: application to polycrystals

A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.

Topology optimization of dissipative metamaterials at finite strains based on nonlinear homogenization

This study presents a novel computational framework for designing optimal dissipative (damping) metamaterials under time-dependent loading conditions at finite deformations. In this framework, finite strain computational homogenization is integrated with a density-based multimaterial topology optimization. In addition, a thermodynamically consistent finite strain viscoelasticity model is incorporated together with an analytical path-dependent sensitivity analysis. Optimization formulations with and without stiffness and mass constraints are considered, and various new damping metamaterial designs are obtained that combine soft viscoelastic and stiff hyperelastic material phases. Multiscale stability analysis using the Bloch wave analysis and rank-1 convexity checks is also carried out to investigate stability of the optimized designs. Stability analyses demonstrate that the inclusion of voids or soft material phases can make a metamaterial more prone to lose micro and macro-stability. Furthermore, the concept of tunable metamaterials is explored wherein metamaterial’s response is steered towards a stable deformation path by tailoring the design with a preselected micro buckling mode.

Non-reciprocal wave propagation in mechanically-modulated continuous elastic metamaterials.

Acoustic and elastic metamaterials with time- and space-dependent effective material properties have recently received significant attention as a means to induce non-reciprocal wave propagation. Recent analytical models of spring-mass chains have shown that external application of a nonlinear mechanical deformation, when applied on time scales that are slow compared to the characteristic times of propagating linear elastic waves, may induce non-reciprocity via changes in the apparent elastic modulus for perturbations around that deformation. Unfortunately, it is rarely possible to derive analogous analytical models for continuous elastic metamaterials due to complex unit cell geometry. The present work derives and implements a finite element approach to simulate elastic wave propagation in a mechanically-modulated metamaterial. This approach is implemented on a metamaterial supercell to account for the modulation wavelength. The small-on-large approximation is utilized to separate the nonlinear mechanical deformation (the "large" wave) from superimposed linear elastic waves (the "small" waves), which are then analyzed via Bloch wave analysis with a Fourier expansion in the harmonics of the modulation frequency. Results on non-reciprocal wave propagation in a negative stiffness chain, a structure exhibiting large stiffness modulations due to the presence of mechanical instabilities, are then shown as a case example.

Broadside Scanning Asymmetric SIW LWA With Consistent Gain and Reduced Sidelobe

In this paper, a backward-to-forward continuous beam scanning leaky wave antenna is presented in substrate integrated waveguide technology. The antenna radiates from a continuous longitudinal slot etched on its broad wall. The slot is excited by using periodic H-plane steps. An impedance matched unit cell structure is used to suppress the open stopband in the broadside direction. Bloch wave analysis is used to obtain the propagation characteristics of the antenna. However, due to radiation being obtained only from the shunt-type radiating element, namely, the longitudinal slot, a gain dip is observed in the broadside direction. Transversal asymmetry is then introduced in the structure to eliminate the gain dip and obtain the consistent gain. A prototype of the antenna is fabricated and measured. Continuous beam scanning is achieved from -29 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sup> to +30 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sup> about the broadside direction with a gain variation of less than 2 dB over 8.0-12.4 GHz. A measured peak gain of 16.1 dBi is obtained. Key structural design parameters of the antenna are identified for controlling the leakage rate. Next, Taylor tapered aperture illumination is used to obtain low sidelobe level (SLL). The Taylor tapered antenna is also fabricated with the measured SLL below -21 dB.

Diffusive stability for periodic metric graphs

AbstractWe consider a nonlinear diffusion equation on an infinite periodic metric graph. We prove that the terms which are irrelevant w.r.t. linear diffusion on the real line are irrelevant w.r.t. linear diffusion on the periodic metric graph, too. The proof is based on L1‐ estimates combined with Bloch wave analysis for periodic metric graphs.

The Korteweg–de Vries, Burgers and Whitham limits for a spatially periodic Boussinesq model

We are interested in the Korteweg–de Vries (KdV), Burgers and Whitham limits for a spatially periodic Boussinesq model with non-small contrast. We prove estimates of the relations between the KdV, Burgers and Whitham approximations and the true solutions of the original system that guarantee these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over their natural timescales. The proof is based on Bloch wave analysis and energy estimates and is the first justification result of the KdV, Burgers and Whitham approximations for a dispersive partial differential equation posed in a spatially periodic medium of non-small contrast.

Topological phase transition in mechanical honeycomb lattice

Topological materials provide a new tool to direct wave energy with unprecedented precision and robustness. Three elastic topological phases, the valley Hall, Chern and spin Hall insulators, are currently studied, and they are achieved separately in rather distinct configurations. Here, we explore analytically various topological phase transitions for in-plane elastic wave in a unified mass-spring honeycomb lattice. It is demonstrated that the three elastic topological phases can be realized in this single lattice by designing mass, stiffness or introducing Coriolis’ effect. In particular, the interface between valley Hall and Chern insulators is found to support topological interface mode for the first time. Perturbation method is used to derive the analytic effective continuum model in the neighbor of band degeneracy, and the physics in topological phase transitions are revealed through evaluation of topological invariants. The topologically protected interface states, their decaying profile as well as the pseudo-spin-indicating polarization specific for elastic wave are systematically analyzed, and these results are further confirmed numerically by Bloch wave analysis of domain wall strip and transient simulation of finite sized sample. This study offers a concise and unified analytical model to explore topology nature of elastic wave, and can provide intuitive guidance to design of continuum mechanical topological materials.

Wave Directionality in Three-Dimensional Periodic Lattices

This work focuses on elastic wave propagation in three-dimensional (3D) low-density lattices and explores their wave directionality and energy flow characteristics. In particular, we examine the dynamic response of Kelvin foam, a simple-and framed-cubic lattice, as well as the octet lattice, spanning this way a range of average nodal connectivities and both stretching-and bending-dominated behavior. Bloch wave analysis on unit periodic cells is employed and frequency diagrams are constructed. Our results show that in the low relative-density regime analyzed here, only the framed-cubic lattice displays a complete bandgap in its frequency diagram. New representations of iso-frequency contours and group-velocity plots are introduced to further analyze dispersive behavior, wave directionality, and the presence of partial bandgaps in each lattice. Significant wave beaming is observed for the simple-cubic and octet lattices in the low frequency regime, while Kelvin foam exhibits a nearly isotropic behavior in low frequencies for the first propagating mode. Results of Bloch wave analysis are verified by explicit numerical simulations on finite size domains under a harmonic perturbation.

Formation of local resonance band gaps in finite acoustic metamaterials: A closed-form transfer function model

The objective of this paper is to use transfer functions to comprehend the formation of band gaps in locally resonant acoustic metamaterials. Identifying a recursive approach for any number of serially arranged locally resonant mass in mass cells, a closed form expression for the transfer function is derived. Analysis of the end-to-end transfer function helps identify the fundamental mechanism for the band gap formation in a finite metamaterial. This mechanism includes (a) repeated complex conjugate zeros located at the natural frequency of the individual local resonators, (b) the presence of two poles which flank the band gap, and (c) the absence of poles in the band-gap. Analysis of the finite cell dynamics are compared to the Bloch-wave analysis of infinitely long metamaterials to confirm the theoretical limits of the band gap estimated by the transfer function modeling. The analysis also explains how the band gap evolves as the number of cells in the metamaterial chain increases and highlights how the response varies depending on the chosen sensing location along the length of the metamaterial. The proposed transfer function approach to compute and evaluate band gaps in locally resonant structures provides a framework for the exploitation of control techniques to modify and tune band gaps in finite metamaterial realizations.