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We investigate the effect of hierarchical designs on the bandgap structure of periodic lattice systems with inner resonators. A detailed parameter study reveals various interesting features of structures with two levels of hierarchy as compared with one level systems with identical static mass. In particular: (i) their overall bandwidth is approxim...
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... Recently, the concept of graded design has been introduced into the design of metamaterials, yielding good results in broadening the bandgaps [42] . For example, Liu and Reina [43] proposed a graded hierarchical architecture through the graded concept, which greatly broadened the bandgap width of hierarchical metamaterials. Wang et al. [44] integrated the graded stiffness into metamaterial sandwich beams to improve the bandgap characteristics in a low-frequency range. ...
A novel elastic metamaterial is proposed with the aim of achieving low-frequency broad bandgaps and bandgap regulation. The band structure of the proposed metamaterial is calculated based on the Floquet-Bloch theorem, and the boundary modes of each bandgap are analyzed to understand the effects of each component of the unit cell on the bandgap formation. It is found that the metamaterials with a low elastic modulus of ligaments can generate flexural wave bandgaps below 300 Hz. Multi-frequency vibrations can be suppressed through the selective manipulation of bandgaps. The dual-graded design of metamaterials that can significantly improve the bandgap width is proposed based on parametric studies. A new way that can regulate the bandgap is revealed by studying the graded elastic modulus in the substrate. The results demonstrate that the nonlinear gradient of the elastic modulus in the substrate offers better bandgap performance. Based on these analyses, the proposed elastic metamaterials can pave the way for multi-frequency vibration control, low-frequency bandgap broadening, and bandgap tuning.
... Graded metamaterials, i.e., a quasi-periodic metamaterial achieved by gradually varying the configuration of the unit cell, showed potential in enlarging the bandgap. 31,32 Many researchers considered gradually modifying the geometric/material properties of the local resonators, including the stiffness/mass of mass-spring resonators, 33,34 the height of Helmholtz resonators, 35 and the length of beam-type resonators, 36 to broaden the vibration suppression region of metamaterials. However, it must be mentioned that once these metamaterials are fabricated, it is difficult to modify their grading configuration. ...
This paper presents a new class of graded metamaterial beams by leveraging actively controllable resonators (ACR). The metamaterial comprises a homogeneous host beam that is mounted with negative capacitance shunted piezoelectric cantilever beams, each of which has a tip mass block. Properly changing the negative capacitances (NCs) in the stiffening/softening shunt circuits can control the formed bandgaps, providing greater adjustability and flexibility. Specifically, using modal analysis and considering higher modes of flexural vibrations, the ACR is simplified to an equivalent lumped parameter system with a correction factor applied to the reaction force. We demonstrate the relationship between the derived equivalent parameters of the ACR and NC for different circuitry configurations. A finite element model is built to validate the theoretical models of the ACR and the proposed metamaterial. Subsequently, a grading strategy is proposed to determine the NC values of ACR arrays for achieving broadband vibration suppression. A mechanical damping enhancement phenomenon that can contribute to forming an aggregated band is observed when resistances are introduced into the stiffening circuits. Three circuit configurations are examined, i.e., stiffening, softening, and hybrid circuits. The results showed that a proper grading coefficient can effectively suppress broadband vibration in the low-frequency range.
... The multi-component structures can produce richer band gaps in the target frequency range, this brings more possibilities for band gap regulation and widening, but the band gap mechanism is consistent with the three-component structure, which essentially adds multiple resonant units on the basis of the three-component structure. Most multi-component structures are implemented by constantly nesting coating layers and scatterers [40][41][42][43][44], the so-called inward hierarchical structures. However, as the number of layer increases, this approach will lead to weaker structural stability due to the differences in material properties of the coating layer and scatterer. ...
The local resonance phononic crystals have advantages in obtaining low frequency band gaps, while the local resonance band gap obtained by traditional structures is inconvenient for engineering applications due to its narrow width. In this manuscript, a theoretical framework for the design and analysis of the Outward Hierarchical Local Resonance Euler Beam (OHLREB) structure has been introduced, which considers the foundational constraints, non-rigid interfaces, and the damping characteristics of the coating layer, leading to a significant enhancement in the low-frequency band gap properties. To analyze the band gap properties of OHLREB, the improved transfer matrix method is derived to obtain the band structure, and the spectral element method is applied to calculate the frequency response function. Furthermore, the band gap mechanism is investigated based on modal analysis of primitive cell and the time domain analysis and energy period analysis of OHLREB structure. The results indicate that improved transfer matrix method can offer a more streamlined and transparent process for solving eigenvalues and determining attenuation intervals, which can be used for the theoretical analysis of band gap regulation. It overcomes the constraints of traditional approaches by reducing the computational burden associated with matrix inversions and multiplications, thereby enhancing the efficiency of band gap analysis. The band gap mechanism investigation demonstrates that the energy conversion period is considerably longer than the load period in band gap frequency, leading to multiple conversions of the kinetic energy of the scatterer and the elastic strain energy of the coating layer within a single energy cycle, which leads to the strong local resonance and results in the attenuation of elastic waves in the band gap range. The parameter analysis is carried out to reveal the regulation laws of band gap, the results reveal that band gap intervals and attenuation efficiency can be optimized to obtain low frequency and broadband by configurating properly the control parameters, such as the matrix reaction modulus, interface stiffness, the coating layer damping, and the scatterer combination. Based on the analysis results, the design and control directions for obtaining the wider and lower frequency band gaps are given. The conclusions obtained in this manuscript can provide theoretical support and practical design perspectives for multi-frequency vibration control field.
... -mass metamaterial model. Such extended models have been considered in Fang et al. (2017), Zhou et al. (2015), Oyelade and Akano (2020), Bukhari and Barry (2020), Liu and Reina (2018), Huang and Sun (2010), Hu et al. (2017). A nonlinearly elastic chain with masses when attached mass with are attached through two main masse has been studied in Fang et al. (2017). ...
Three nonlinearly elastic extended mass-in-mass metamaterial models are asymptotically studied using their long wavelength continuum limits. The governing nonlinear equations with dispersion are obtained for longitudinal strain including dispersion effects. Both nonlinear stiffness of the main chain spring and those of the attached masses are taken into account. It is shown that the nonlinearities and dispersion differently affect the dynamics of the longitudinal strain waves for the models considered.
... One effective way to realize the elastic wave metamaterials is to design the artificial periodic structures with local resonators [12][13][14][15]. From the investigations of wave propagation in lattice systems consisting of massin-mass units, people can observe frequency stop bands near the local resonance frequency in which spatial decay of wave amplitude occurs [16,17]. As a result, this kind of metamaterials shows potential applications in vibration isolation. ...
In this study, the dynamic response of a growing inclusion in an elastic wave metamaterial with local resonators is investigated. It is assumed that the inclusion at a constant speed is composed of particles with larger or smaller mass than the surrounding lattice is. Based on the integral transform, wave equations of the growing inclusion are derived as the Wiener–Hopf form. In order to show the effect of local resonators, stop band properties and the ratio of the tip to end displacements of the inclusion are considered. Numerical results show that the local resonators can make the displacement ratio tend to the case without inclusion. It indicates that the outstanding ability of elastic wave metamaterials inhibits structural deformation and defect growth.
... In recent years, several researchers propose hierarchical geometries for LRMs to obtain wider band gaps [3,4]. Numerical simulations of a continuum system exhibiting a hierarchical geometry may represent a huge computational burden and analytical predictions of their dynamical properties are mainly restricted to discrete systems of lumped masses and springs. ...
Local resonant metamaterials are a class of microstructured man-made material which attenuate the propagation of waves in certain frequency ranges, known as band gaps. In this work, we study through asymptotic homogenization the anti-plane shear wave propagation in metamaterial with a stiff matrix and soft inclusions, periodically distributed, which present a hierarchical geometry. Band gaps of the metamaterial are then analytically predicted by the intervals of frequency in which the effective mass becomes negative.
... In the optics and acoustic fields, graded structures had been used to broaden bandgaps in many metamaterials. [45][46][47] Introducing the concept of a graded configuration to LRSMs may solve the bandwidth problem. Furthermore, we consider the availability of the composites and the ease of realization of the configurations. ...
By introducing the concept of a graded structure to seismic metamaterials, a new type of graded seismic metamaterial assembled using four steel sections with different graded levels is proposed to investigate its attenuation performance for surface waves. The dispersion curves and vibration modes are obtained using the finite element method and the sound cone method. A comparative analysis of the band gap characteristics of the four graded seismic metamaterials shows that an increase of the graded level is beneficial for widening the total band gap to a much larger relative bandwidth in the range of 0.1–13.07 Hz. In addition, a detailed analysis of the vibration modes reveals that local resonance is the main mechanism for the generation and change of the three band gaps. Moreover, the filling materials in the cavities, material and geometric parameters of the structure play important roles in the distribution and relative bandwidth of the band gaps. Finally, frequency–domain analysis is carried out on a finite system, and the agreement with the bandgaps is verified. This study paves the way for the design of graded seismic metamaterials. This concept allows flexible manipulation of the surface wave propagation by adjusting the graded level, fillers, geometric parameters of the steel sections, and soil materials to achieve seismic wave attenuation in low-frequency broadband.
... Functionally Graded Materials (FGMs) represents a kind of new type high-performance composites with gradually varying material contents [15][16][17][18][19][20], which gives us insights into the design of broadband PnCs. The graded design is currently applied to the local resonant phononic crystals for the purpose of broadening the band gap [21][22][23][24][25][26], while less research has been done on Bragg scattering phononic. Kushwaha [27] first achieved ultra-wideband filter of noise by placing five groups of periodic metal rods in series, so that the band gaps of each group of structures are superimposed on each other. ...
A new phononic crystal with graded supercell configuration is proposed to broaden the Bragg scattering band gaps. The proposed phononic crystal is made up of a periodic arrangement of supercells, and the supercells are composed of unit cells with graded structural parameters. The mechanical model of the graded phononic crystals is established based on transfer matrix method to investigate in-plane elastic waves propagating and band structures of the periodic system. Numerical results show that the graded structural design can merge adjacent multiple band gaps into an extremely broad one. Modal analysis shows that the mechanism of band gap broadening is that the graded supercell configuration breaks some symmetries of the phononic crystal, resulting in the opening of Dirac cone and creation of new band gaps. The effects of the main structural parameters related to graded supercell design on band gap broadening are studied by simulation and verified by experiment. The present study is beneficial to the design of new functional materials with broadband vibration isolation performance.
... Metamaterials are architectured materials whose mechanical properties go beyond those of classical materials thanks to their heterogeneous microstructure. This allows them to show exceptional static/dynamic features such as negative Poisson's ratio [18], twist in response to being pushed or pulled [12,32], band-gaps [7,20,30,42], cloaking [6,25], focusing [3,13], channeling [11,14], negative refraction [4,15,19,44,47], etc. ...
In this paper, we present a unit cell showing a band-gap in the lower acoustic domain. The corresponding metamaterial is made up of a periodic arrangement of one unit cell. We rigorously show that the relaxed micromorphic model can be used for metamaterials’ design at large scales as soon as sufficiently large specimens are considered. We manufacture the metamaterial via metal etching procedures applied to a titanium plate so as to show that its production for realistic applications is viable. Experimental tests are also carried out confirming that the metamaterials’ response is in good agreement with the theoretical design. In order to show that our micromorphic model opens unprecedented possibilities in metastructural design, we conceive a finite-size structure that is able to focus elastic energy in a confined region, thus enabling its possible subsequent use for optimizing complex structures. Indeed, thanks to the introduction of a well-posed set of micromorphic boundary conditions, we can combine different metamaterials and classical Cauchy materials in such a way that the elastic energy produced by a source of vibrations is focused in specific collection points. The design of this structure would have not been otherwise possible (via e.g., direct simulations), due to the large dimensions of the metastructure, couting hundreds of unit cells.
... Beyond the elastic and acoustic waves, Lu et al. have studied "rainbow" effect of optical wave in graded nanostructures [54,55] in both simulation and experiment, and they have given review of topology rainbow [56]. In effect, the "rainbow" effect of different waves through graded structures without the topology [57,58] has long caused attention for the purpose of trapping wave energy [59][60][61], broadening band gap [62], and guiding elastic wave [63]. Yet to now, the experimental observation of elastic wave propagation through graded topological interface is seldom reported for plate wave. ...
In this work, we construct on-chip valley phononic crystal plates by building up triangular silicon pillars on the silicon plate. We introduce the graded interface sliding for two typical interfaces ψBA and ψAB, by modulating the horizontal lattice constant in a selected region in PC plate. Based on simulation and laser ultrasonic technique, we present the frequency range evolution of topological edge states from the gapless feature to gapped one. We demonstrate the displacement distribution of topological edge states in two graded interfaces, magnifying numerically the topological rainbow phenomena. In particular, we characterize the plate wave propagation through both types of graded interfaces, observe the abnormal refraction pattern behind the truncated interfaces, and underline abnormal refraction to the splitting of wave vector in PC plate. Our graded interfaces provide new clue to control elastic wave propagation based on valley degree of freedom.
