Finite Crystal Research Articles

Pseudospin induced topological corner state at intersecting sonic lattices

Inspired by the discoveries of electronic topological phases and topological insulators, topologically protected boundary states in classical wave-based systems have attracted considerable interest in the last decade. Most recently, acoustic higher-order topological insulators and Kekul\'e-distorted sonic lattices have been proposed to support topological corner states and zero-dimensional bound states. Here, we demonstrate a domain wall induced topological corner state that is bound at the crossing point among finite acoustic graphenelike crystals. The approach is based on designing multipolar pseudospin resonances, which give rise to topologically trivial and nontrivial transitions across the domain walls that flank this unusual corner excitation toward the crossing point. By deliberately adding a substantial amount of defects into the cavities of the sonic lattice, we find that the pseudospin induced topological corner state remains entirely unaffected and pinned spectrally to the complete audible band gap. Our findings may thus have the potential to broaden the possibilities for sound confinement and focusing.

Influence of Lateral Constraints on Wave Propagation in Finite Granular Crystals

Abstract In the presented work, wave dynamics of 2D finite granular crystals of polyurethane cylinders under low-velocity impact loading was investigated to gain better understanding of the influence of lateral constraints. The deformation of the individual grains in the granular crystals during the impact loading was recorded by a high-speed camera and digital image correlation (DIC) was used to calculate high fidelity kinematic and strain fields in each grain. These grain-scale kinematic and strain fields were utilized for the computation of the intergranular forces at each contact using a granular element method (GEM) based mathematical framework. Since the polyurethane were viscoelastic in nature, the viscoelasticity constitutive law was implemented in the GEM framework and it was shown that linear elasticity using the strain rate-dependent coefficient of elasticity is sufficient to use instead of a viscoelastic framework. These particle-scale kinematic and strain field measurements in conjunction with the interparticle forces also provided some interesting insight into the directional dependence of the wave scattering and attenuation in finite granular crystals. The directional nature of the wave propagation resulted in strong wave reflection from the walls. It was also noteworthy that the two reflected waves from the two opposite sidewalls result in destructive interference. These lateral constraints at different depths leads to significant differences in wave attenuation characteristics and the finite granular crystals can be divided into two regions: upper region, with exponential wave decay rate, and lower region, with higher decay rate.

A comparative study of integration algorithms for finite single crystal (visco-)plasticity

Computational aspects of finite crystal plasticity have been discussed in the literature for some time, yet open challenges regarding the robustness and uniqueness of predictions for different algorithms remain. In this contribution, a careful comparison between different formulations of single crystal plasticity is therefore presented, covering both rate-dependent and rate-independent behavior in the finite deformation setting. To this end, algorithms are developed for an augmented-Lagrangian formulation and a formulation based on nonlinear complementary functions in the rate-independent case. For the rate-dependent case, algorithmic treatments are considered for two commonly employed viscoplastic formulations. The robustness of the different algorithms is assessed by means of simulations at the material point level, involving proportional fully deformation-controlled tests and non-proportional loading cycles in stress space. The augmented-Lagrangian formulation and a particular viscoplastic formulation are found to be superior in terms of robustness compared to the other algorithm considered in their respective category. It is then further observed that these algorithms of choice, which perform well for the ideally-plastic case and for Taylor-type hardening, induce a significant reduction in acceptable deformation increment size for an anisotropic, phenomenological hardening law, due to the rather complex determination of the active slip systems.Additionally, it is found from a sensitivity analysis that increasing the degree of anisotropy in the hardening law amplifies the sensitivity of the model’s response to small initial misorientations, both in the rate-dependent and the rate-independent case. A sensitivity analysis is also applied in the ideally-plastic case to demonstrate that for common choices of viscous parameters, the response of a particular rate-dependent formulation of the crystal plasticity model not necessarily possesses the same characteristic features as the response obtained by the corresponding rate-independent augmented-Lagrangian formulation. Therefore, it is recommended to apply the augmented-Lagrangian formulation to determine the rate-independent model response, rather than employing a rate-dependent formulation with limiting values of the viscous parameters due to the superior robustness of the former.The application of the augmented-Lagrangian formulation in the solution of an inhomogeneous fixed-end torsion problem by means of finite element analysis demonstrates the beneficial numerical properties of this formulation and the ability of the material model to reproduce characteristic features observed in experiments on single crystal copper samples.

Study on the vibration attenuation property of one finite and hybrid piezoelectric phononic crystal beam

In this paper, a theoretical method is proposed to effectively improve the vibration attenuation characteristics of a finite hybrid piezoelectric phononic crystal (PC) beam utilizing the non-uniform distribution of shunting circuits. Considering the damping of resonators, the vibration transmissibility method (TM) for the finite electromechanical system is derived based on the Timoshenko beam theory. Numerical results are validated by the finite element method (FEM). Subsequently, a comprehensive study is conducted to investigate the influences of the structural and electrical parameters on the vibration attenuation property of the uniform PC. Furthermore, the organized disorder of the shunting circuits is introduced into the system. Combining with various types of non-uniform circuit configurations, the variation of the vibration attenuation is also examined. It is demonstrated that the hybrid PCs offer more tunable mechanisms to design target band-gaps, compared with the purely mechanical and piezoelectric PCs. The low-frequency vibration reduction over a broadband range could be achieved through the selective coupling ways of the mechanical local resonant band-gap (LRG), electromechanical LRG and Bragg band-gap (BG). It is expected to enhance the vibration attenuation and availability of hybrid PCs using the suitable distribution of shunting circuits in practice.

Analysis of scattering by finite sonic crystals in free field with infinite elements and normal modes

Acoustic exterior problems are in this paper solved numerically using the Astley-Leis infinite element method (IFEM). Normal modes can be determined thanks to the frequency independence of the system matrices. Convergence properties of the harmonic sound pressure solution as well as of normal mode eigenvalues are investigated for two-dimensional elliptical computational domains to estimate the essential requirements of accuracy. A relationship between the half-axis ratio of the ellipses and the eigenvalues is identified. By solving half-space problems, symmetry of the computational domains is utilized, which is shown for the first time for frequency-independent normal modes in exterior acoustics. This paper discusses applicability of the modes in the example of sonic crystals—periodic arrays of scatterers, in this context denoted as acoustic meta-atoms—that have recently attracted attention for their possible use as noise barriers. It can be shown that the sound-insulating effect of finite sonic crystals and individual meta-atoms in the free field can be related to normal modes in exterior acoustics. With the help of this approach, the absorption by Helmholtz resonators and due to a boundary admittance inside are studied. This work provides a new point of view and physical insights into the effects and underlying physics of sound insulation by finite sonic crystals and acoustic meta-atoms in free field. Although it is not the intention of this article to optimize the arrays, the method of normal modes in exterior acoustics is presented as an appropriate and novel tool for their dimensioning and design.

Bistability phenomena based on frequency-sensitive self-collimation in nonlinear photonic crystals

By introducing the change of beam behavior near the frequency sensitive self-collimation point, a new kind of bistability is observed in a finite nonlinear photonic crystal (PhC). The mechanism of such a PhC bistability is from the complex interplay between cavity resonance, nonlinearity and beam behavior. Additionally, it has unique properties, including: high frequency sensitivity, small volume, short switching time, robustness and position flexibility. With the consideration of beam behavior, a new theory based on the coupled-mode equation has been constructed, whose result agrees with the numerical one very well. The new PhC bistability could be widely used to design various optical devices, as a bistable all-optical switch has also been demonstrated.

Finite Crystallization and Wulff shape emergence for ionic compounds in the square lattice

We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational energies featuring two-body short-ranged particle interactions which favor some reference distance between different atomic types and contain repulsive contributions for atoms of the same type. We first prove that ground states are connected subsets of the square lattice with alternating arrangement of the two atomic types in the crystal lattice, and address the emergence of a square macroscopic Wulff shape for an increasing number of particles. We then analyze the signed difference of the number of the two atomic types, the so-called net charge, for which we prove the sharp scaling in terms of the particle number n. Afterwards, we investigate the model under prescribed net charge. We provide a characterization for the minimal energy and identify a critical net charge beyond which crystallization in the square lattice fails. Finally, for this specific net charge we prove a crystallization result and identify a diamond-like Wulff-shape of energy minimizers which illustrates the sensitivity of the macroscopic geometry on the net charge.

Soliton lattices originating from excitons interacting with high-intensity fields in finite molecular crystals

The effects of long-range intermolecular interactions on characteristic features of soliton bound states, consisting of localized excitons and polaritons in molecular crystals interacting with a high-intensity optical field, are investigated. Analytical solutions to the resulting modified nonlinear Schr\"odinger equation are obtained in terms of elliptic-type bright- and dark-soliton structures, which are assumed to correspond to periodic trains of pulse solitons and kink solitons respectively. Long-range intermolecular interactions are shown to renormalize the exciton-polariton interaction strength, hence generating a significantly huge increase in amplitudes of the bright solitons but a decrease in amplitudes of dark solitons. Results suggest that long-range intermolecular interactions hold relevant roles, both qualitatively and quantitatively, in the formation of amplitude and phase modulated strongly nonlinear exciton-polariton solitary-wave patterns, as well as in the energy transfer along molecular crystals interacting with high-intensity optical fields.

On the robustness of stress integration algorithms for finite single crystal (visco)‐plasticity

AbstractThe accurate determination of the slip activity in single crystal plasticity models is crucial to predict the mechanical response and the reorientation of the crystal. Therefore, a comparison of two rate‐independent and two rate‐dependent formulations of single crystal plasticity is carried out, subject to homogeneous proportional and non‐proportional deformation histories. From the simulation results it is concluded that the robustness of the corresponding stress‐integration algorithm depends on the complexity of the hardening law.

Experimental investigation of test particle induced micro-structural changes in a finite two-dimensional complex plasma crystal

The microstructural changes induced in a two dimensional complex plasma crystal by the movement of a single extra particle existing below it are investigated experimentally using a DC glow discharge device. The finite monolayer crystal consisting of monodispersive melamine formaldehyde particles is created in the background of an Argon plasma. The crystal formation is frequently found to be accompanied by the presence of one or more slightly heavier particles suspended a little below the monolayer. The interplay of one such test particle with the crystal is investigated for two distinct cases—(i) when the particle remains confined (trapped) in the space below the crystal and (ii) when it interacts for a short while with the crystal and then moves out of the vicinity. The trapped particle orbit induces permanent structural changes in the crystal in the form of microcracks, and these changes are identified using a variety of diagnostic analyses. The cracks can be enhanced by energizing the test particle with an incident laser. The passing particle, on the other hand, induces only transient changes in the crystal from which it is able to recover and return to its initial state in a clear demonstration of the viscoelastic property of a strongly coupled system.

One-dimensional multiperiodic photonic structures: A new route in photonics (four-component media)

We study the transmittivity spectra of one-dimensional finite three-periodic photonic crystals of the structure [(ab)N(cd)M]K composed of four different layers a, b, c, and d being dielectric oxides Al2O3, SiO2, TiO2, and ZrO2, respectively. We analyze modification of the transmittivity spectra of near-infrared electromagnetic waves of the TE and TM polarizations in the vicinity of the photonic bandgaps with variations of the sub-cell numbers N and M and super-cell number K, the incidence angle, and the order of the layers in the structure. We propose the classification of three-periodic photonic crystals by the magnitude and sign of the optical contrast between the layers in pairs (ab) and (cd) forming the sub-cells and discuss common spectral properties as well as their differences in these structures.

Nonspecular Effects That Occur as a Quasi-Plane Wave Passes through a Finite Photonic Crystal with an Asymmetric Polariton Spectrum

First-order nonspecular effects emerge in the plane of layers for both reflected and transmitted waves (particularly the spatial and angular Goos–Hanchen effect) for a quasi-plane of the TM (or TE) type normally incident from outside on a finite one-dimensional magnetic photonic crystal with a spectrum of normal magnetic polaritons, asymmetrical with respect to the inversion of the direction of wave propagation.

Photonic crystal-based compact high-power vacuum electronic devices

This paper considers how the finite dimensions of a photonic crystal placed inside a resonator or waveguide affect the law of electron beam instability. The dispersion equations describing e-beam instability in the finite photonic crystal placed inside the resonator or waveguide (a bounded photonic crystal) are obtained. Two cases are considered: the conventionally considered case, when diffraction is suppressed, and the case of direct and diffracted waves having almost equal amplitudes. The instability law is shown to be responsible for increase of increment of instability and decrease of length, at which instability develops, for the case when amplitude of diffracted wave is comparable with that of direct one, that happens in the vicinity of $\pi$-point of dispersion curve. Application of photonic crystals for development of THz sources at electron beam current densities available at modern accelerators is discussed.

Topological pumping of edge states via adiabatic passage

Topological pumping of edge states in finite crystals or quasicrystals with non-trivial topological phases provides a powerful means for robust excitation transfer. In most schemes of topological pumping, the edge states become delocalized and immersed into the continuum during the adiabatic cycle, requiring extremely slow evolution to avoid nonadiabatic effects. Here a scheme of topological pumping based on adiabatic passage of edge and interface states is proposed, which is more robust against nonadiabatic effects and avoids delocalization of the state over the entire adiabatic cycle. The scheme is illustrated by considering adiabatic passage in a system sustaining two topologically-protected edge states and one interface state, realized by interfacing two dimerized Su-Schrieffer-Heeger chains with different topological order

Gradient index phononic crystals and metamaterials

Abstract Phononic crystals and acoustic metamaterials are periodic structures whose effective properties can be tailored at will to achieve extreme control on wave propagation. Their refractive index is obtained from the homogenization of the infinite periodic system, but it is possible to locally change the properties of a finite crystal in such a way that it results in an effective gradient of the refractive index. In such case the propagation of waves can be accurately described by means of ray theory, and different refractive devices can be designed in the framework of wave propagation in inhomogeneous media. In this paper we review the different devices that have been studied for the control of both bulk and guided acoustic waves based on graded phononic crystals.

Thermal echo in a finite one-dimensional harmonic crystal

An instant homogeneous thermal perturbation in the finite harmonic one-dimensional crystal is studied. Previously it was shown that for the same problem in the infinite crystal the kinetic temperature oscillates with decreasing amplitude described by the Bessel function of the first kind. In the present paper it is shown that in the finite crystal this behavior is observed only until a certain period of time when a sharp increase of the oscillation amplitude is realized. This phenomenon, further referred to as the thermal echo, occurs periodically, with the period proportional to the crystal length. The amplitude for each subsequent echo is lower than for the previous one. It is obtained analytically that the time-dependence of the kinetic temperature can be described by an infinite sum of the Bessel functions with multiple indices. It is also shown that the thermal echo in the thermodynamic limit is described by the Airy function.

The acoustic insulation performance of infinite and finite locally resonant metamaterial and phononic crystal plates

The bending wave stop band behavior of locally resonant metamaterial and phononic crystal plates can strongly improve the vibration performance of flexible plates in targeted frequency ranges. Their acoustic insulation performance, however, is found to strongly depend on the stop band mechanism, also influencing the frequency range of improvement. While sound transmission loss improvements are predicted for both infinite periodic plates, the finite phononic crystal plate performance can differ significantly.

Electromagnetic density of modes in one-dimensional photonic crystals containing dispersive metamaterials

The electromagnetic density of modes (DOM) in a finite one-dimensional photonic crystal containing dispersive metamaterials is computed using Wigner’s time approach. The expression of DOM is derived using the transfer matrix method. Different structural parameters, such as relative thickness, incident angle, and total number of unit cells are varied and their effects are investigated. It is observed that relative thickness, angle of incidence, and total number of unit cells all play an important role in determining the properties of DOM depending on the frequency regime under consideration.

Peculiarities of electromagnetic field distribution in a finite one-dimensional photonic crystal at oblique incidence of radiation

The effect of increasing an electric field E is investigated for oblique incidence of radiation on a finite one-dimensional photonic crystal. The peculiarities in the field distribution in the structure E(x) are found to be due to the interference of counterpropagating Bloch waves; therefore, the conditions for increasing the field to a maximum value |E|max coincide with those for achieving the maximum band gap Eg of an infinite crystal. It is shown that a significant (several times for structures with 30 periods) growth of |E|max is observed with increasing angle of incidence θ for wavelengths λ, which correspond to transmission maxima T nearest to the band gap boundaries. The increased interference of the counterpropagating Bloch modes is caused by an enhancement of the x-component contrast (perpendicular to the planes of the layers) of the wave numbers k1x/k2x in the structure layers. Along with this, there also arise characteristic features in the distribution of E(x) associated with the vanishing of the forbidden band width. In this case, the field distribution in all periods is the same. Two types of such features are observed in the case of oblique incidence. The former takes place for p-polarisation at an angle of incidence θ, which is an analogue of the Brewster angle. The latter corresponds to the situation when the length of each structure layer is equal to an integer of half-waves. This occurs simultaneously for s- and p-polarisations and is observed in a limited range of layer widths only for the second and highest forbidden bands.

Band-gap and pass-band classification for oblique waves propagating in a three-dimensional layered functionally graded piezoelectric phononic crystal

Three-dimensional time-harmonic wave motion in a layered functionally graded (FG) piezoelectric periodic composite (phononic crystal) composed of a finite or an infinite number of unit-cells is considered. A longitudinal or transverse plane waves incident obliquely to the interfaces of a finite phononic crystal between two half-spaces is studied. The paper proposes a semi-analytical method to simulate and analyse the wave fields in a phononic crystal in the case of arbitrary angles of incidence. It is shown that the method is numerically stable for an arbitrary number of unit-cells in finite phononic crystals. Several kinds of pass-bands and band-gaps can be distinguished by employing the derived semi-analytical expressions: band-gaps, pass-bands, low transmission pass-bands, quasi-longitudinal and quasi-transverse band-gaps. Using the present approach a detailed parametric analysis of the influences of the type and incidence angle of an incident wave, and the material and geometrical parameters of the FG interlayers on wave propagation is conducted.