Wave propagation in fractal-inspired self-similar beam lattices

Lim, Qi Jian; Wang, Pai; Koh, Soo Jin Adrian; Khoo, Eng Huat; Bertoldi, Katia

doi:10.1063/1.4936564

Cited by 59 publications

(33 citation statements)

References 44 publications

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“…• Floquet-Bloch boundary conditions 4 (1) = u 2 (0) e i K 2 /α , v 4 (1) = v 2 (0) e i K 2 /α , v 4 (1) = v 2 (0) e i K 2 /α u 4 (1) = u 2 (0) e i K 2 /α , v 4 (1) +ω 2 1 α 2 v 4 (1) = v 2 (0) +ω 2 1 α 2 v 2 (0) e i K 2 /α , v 4 (1) = v 2 (0) e i K 2 /α ,…”

Section: Introductionunclassified

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Bordiga

Cabras

Bigoni

2019

International Journal of Solids and Structures

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In-plane wave propagation in a periodic rectangular grid beam structure, which includes rotational inertia (so-called 'Rayleigh beams'), is analyzed both with a Floquet-Bloch exact formulation for free oscillations and with a numerical treatment (developed with PML absorbing boundary conditions) for forced vibrations (including Fourier representation and energy flux evaluations), induced by a concentrated force or moment. A complex interplay is observed between axial and flexural vibrations (not found in the common idealization of out-of-plane motion), giving rise to several forms of vibration localization: 'X-', 'cross-' and 'star-' shaped, and channel propagation. These localizations are triggered by several factors, including rotational inertia and slenderness of the beams and the type of forcing source (concentrated force or moment). Although the considered grid of beams introduces an orthotropy in the mechanical response, a surprising 'isotropization' of the vibration is observed at special frequencies. Moreover, rotational inertia is shown to 'sharpen' degeneracies related to Dirac cones (which become more pronounced when the aspect ratio of the grid is increased), while the slenderness can be tuned to achieve a perfectly flat band in the dispersion diagram. The obtained results can be exploited in the realization of metamaterials designed to control wave propagation. and the solution techniques are well-known, many interesting features still remain to be explored. This exploration is provided in the present article, where an exact Floquet-Bloch analysis is performed and complemented with a numerical treatment of the forced vibrations induced by the application of a concentrated force or moment, including presentation of the Fourier transform and energy flow (treated in [26] for free vibrations). It is shown that (i.) aspect ratio of the grid, (ii.) slenderness and (iii.) rotational inertia of the beams decide the emergence of several forms of highly-localized waveforms, namely, 'channel propagation', 'X-', 'cross-', 'star-' shaped vibration modes. Moreover, these mechanical properties of the grid can be designed to obtain flat bands and degeneracies related to Dirac cones in the dispersion diagram and directional anisotropy or, surprisingly, dynamic 'isotropization', for which waves propagate in a square lattice with the polar symmetry characterizing propagation in an isotropic medium.The presented results open the way to the design of vibrating devices with engineered properties, to achieve control of elastic wave propagation.2 In-plane Floquet-Bloch waves in a rectangular grid of beams An infinite lattice of Rayleigh beams is considered, periodically arranged in a rectangular geometry as shown in Fig. 1a, together with the unit cell, Fig. 1b.

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Section: Introductionunclassified

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Bordiga

Cabras

Bigoni

2019

International Journal of Solids and Structures

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“…Recent work has also examined the use of cut hinges topologies in a hierarchical tessellation both from the static mechanical and in-plane wave propagation behaviour [29]. Waves in self-similar domains have several appealing features, like localization phenomena in fluid-filled periodic fractal inclusion acoustic band gap crystals or filters [30], Sierpinsky or quasi-fractal arrangements [31], and the creation of large bandgaps at lower dimensionless frequencies in beam lattices [32]. Reference [29] suggests that the general use of perforations could constitute a quite interesting strategy to design extremely tailorable bandgap materials especially at lower frequencies, due to the ease of producing these 2D metamaterials by simple automatic cutting/CNC machining.…”

Section: Introductionmentioning

confidence: 99%

Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials

Billon

Zampetakis

Scarpa

et al. 2017

Composite Structures

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We describe in this work a composite metamaterial with a hierarchical topology made by tessellating perforations that exhibit an auxetic (negative Poisson's ratio) behaviour. We perform an analysis of the hierarchical structure by evaluating the fractal order of the topologies associated to the perforared composites.The periodic hierarchical lattice configuration shows negative Poisson's ratio characteristics at higher levels of hierarchy, even when the baseline configuration has a topology not exhibiting an auxetic behaviour. We investigate the wave propagation characteristics of these particular hierarchical lattices by using a Bloch Wave approach applied to detailed Finite Element geometries of the unit cell configurations. We show that the level of

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“…Hierarchical structures are composed of structural elements witch themselves have structure. [8][9][10][11] Auxetic materials are materials with a negative Poisson's ratio. For conventional materials, during an uniaxial tensile test, a reduction of the dimensions perpendicular of the load is observed, in the case of auxetic materials an increasing of these dimensions is obtained.…”

Section: Introductionmentioning

confidence: 99%

Parametric study of wave propagation in hierarchical auxetic perforated metamaterials

et al. 2016

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The understanding of wave propagation in a metamaterial with hierarchical, auxetic rectangular perforations is presented in this work. The metamaterial is a 2D structure with chaining horizontal and vertical perforations exhibing auxetic in-plane behaviour. The unit cell of this lattice is identified as the reference level 0. Hierarchical structures are composed of structural elements which themselves have structure. At level 0, 4 rigid squares are present in the unit cell. In each square, the reference structure is used by applying a scale ratio to obtain the level 1. The same strategy is used to reach the upper level in each subunit. A geometric parametric investigation of these rectangular perforations using a numerical asymptotic homogenisation finite element approach is done. Some numerical eigenvalue tools are used for the dispersion analysis of this structure. It is first observed that the total width of Band gaps increases with the hierarchy. The porosity induced by the perforations is taken into account in the mechanical properties. The symmetry of the geometry in the x-y plane allow to define the entire geometry of the unit cell using only 2 parameters: the void aspect ratio, the intercell spacing and the hierarchy level. When decreasing the intercell spacing, the total width of Band gaps increases and the effective stiffness in x and y directions decrease, allowing for increased rotations of the rigid sqares, so auxetic behaviour is greater. Hierarchical levels shift from isotropic to orthotropic, hierarchical levels are always auxetic.

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Wave propagation in fractal-inspired self-similar beam lattices

Cited by 59 publications

References 44 publications

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials

Parametric study of wave propagation in hierarchical auxetic perforated metamaterials

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