Transformation cloaking and radial approximations for flexural waves in elastic plates

Brun, Michele; Colquitt, Daniel; Jones, I. S.; Movchan, A. B.; Movchan, N. V.

doi:10.1088/1367-2630/16/9/093020

Cited by 42 publications

(35 citation statements)

References 40 publications

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“…This flexural systems may show many of the typical anisotropic effects from photonics such as ultrarefraction, negative refraction and Dirac-like cones [6][7][8][9][10]. Structured plates may also show the capability of cloaking applications [11][12][13][14] as a result of inhomogeneous and anisotropic constitutive properties and axial prestress [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Platonic crystal with low-frequency locally-resonant spiral structures: wave trapping, transmission amplification, shielding and edge waves

Morvaridi

Carta

Brun

2018

Journal of the Mechanics and Physics of Solids

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We propose a new type of platonic crystal. The proposed microstructured plate includes snail resonators with low-frequency resonant vibrations. The particular dynamic effect of the resonators are highlighted by a comparative analysis of dispersion properties of homogeneous and perforated plates. Analytical and numerical estimates of classes of standing waves are given and the analysis on a macrocell shows the possibility to obtain localization, wave trapping and edge waves. Applications include transmission amplification within two plates separated by a small ligament. Finally we proposed a design procedure to suppress low frequency flexural vibration in an elongated plate implementing a by-pass system rerouting waves within the mechanical system. Dirac cones in platonic crystals equations, associated respectively to the presence of propagating and evanescent waves. Such waves can be coupled via the boundary or interface contact conditions. In most configurations the flexural waves are led by their Helmholtz component [5] and the homogenized equation can be of parabolic type at special frequencies [9,10]. However, short range wave scattering and Bragg resonance can be strongly influenced by the evanescent waves.Periodic structures play a major role in this field [18], since they create band gaps. These are frequency ranges where waves cannot propagate through the periodic system leading to possible application as acoustic and mechanical wave filters, vibration isolators, seismic shields. Partial band gap can lead to anisotropic wave response that can be used to obtain focusing and localization [19][20][21] as well as polarization properties [22,23].Two physical mechanisms can open band gaps: Bragg scattering and local resonance [24,25]. Bragg scattering is associated to the generation of band gaps at wavelengths of the same order of the unit cell around frequencies governed by the Bragg condition a = n(λ/2), (n = 1, 2, 3, · · · ), where a is the lattice constant of the periodic system and λ the wavelength [26]. Local resonances are associated to internal resonances due to the microstructures, they can be obtained from array of resonators as suggested in the seminal work [27]. Local resonances open tiny band gaps that can be at low frequencies [28][29][30] and inertial amplification mechanism that can widen stop band intervals have been proposed in [31,32]. The platonic system of snail resonatorsWe consider flexural vibrations in Kirchhoff plates. In the time-harmonic regime the transverse displacement W(x) satisfy the fourth-order biharmonic equation

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Section: Introductionmentioning

confidence: 99%

Platonic crystal with low-frequency locally-resonant spiral structures: wave trapping, transmission amplification, shielding and edge waves

Morvaridi

Carta

Brun

2018

Journal of the Mechanics and Physics of Solids

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“…Published in Proceedings of the Royal Society A (2019), 475, 20190283 doi: 10.1098/rspa.2019.0283 the thickness ratio is 1.5 [26] and 0.5 [27] in the only few experimented cases, and in theoretical treatments 5, 1.6 [20], 1.17 [25], and 0.5 [21,35,18]. (iii.)…”

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confidence: 97%

“…Obviously, the chief characteristic of cloaking is overall scattering suppression, a condition which is often reached only in an approximate sense. Quantitative data on scattering reduction are usually not explicitly reported in elasticity [20,35,25,21,18,26]. A performance of 60% [27] and 55% [31] is reported for flexural waves.…”

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confidence: 99%

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Omnidirectional flexural invisibility of multiple interacting voids in vibrating elastic plates

2019

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In elasticity, the design of a cloaking for an inclusion or a void to leave a vibrational field unperturbed by its presence, so to achieve its invisibility, is a thoroughly analyzed, but still unchallenged, mechanical problem. The 'cloaking transformation' concept, originally developed in electromagnetism and optics, is not directly applicable to elastic waves, displaying a complex vectorial nature. Consequently, all examples of elastic cloaking presented so far involve complex design and thick coating skins. These cloakings often work only for problems of unidirectional propagation, within narrow ranges of frequency, and considering only one colaked object. Here, a new method based on the concept of reinforcement, achieved via elastic stiffening and mass redistribution, is introduced to cloak multiple voids in an elastic plate. This simple technique produces invisibility of the voids to flexural waves within an extremely broad range of frequencies and thus surpassing in many aspects all existing cloaking techniques. The proposed design principle is applicable in mechanical problems ranging from the micro-scale to the scale of civil engineering. For instance, our results show how to design a perforated load-bearing building wall, vibrating during an earthquake exactly as the same wall, but unperforated, a new finding for seismic protection.

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“…Using the framework of transformation optics, Colquitt, et al (2013) presented a detailed analysis of a physically realizable square cloak for acoustic, out-of-plane shear elastic and electromagnetic waves. Brun et al (2014) developed a model of a broadband invisibility cloak for channeling flexural waves in thin plates around finite inclusions. Diatta and Guenneau (2014) have investigated spherical cloaks for solid elastic waves using a radially symmetric linear geometric transform.…”

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Manipulation of the propagation of out-of-plane shear waves

Gao

2015

International Journal of Solids and Structures

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Please cite this article as: Wu, L., Gao, P., Manipulation of the propagation of out-of-plane shear waves, International Journal of Solids and Structures (2015), doi: http://dx.Abstract: For a given elastic medium, there is a one-to-one correspondence relation between material properties (elastic moduli and density) and the propagation path of elastic waves. Based on this idea, we propose a new method for designing a cylindrical cloak invisible to out-of-plane shear waves. We begin by writing the Hamiltonian governing the path trajectory of out-of-plane shear waves in an inhomogeneous orthotropic medium. Based on Hamilton's equations of motion, we derive the ray equation for out-of-plane shear waves and the differential equation that describes the optimal spatial distribution of material properties in the cylindrical cloak.To solve these two differential equations, two boundary conditions are imposed in terms of the continuity conditions of displacement and traction on the outer boundary of the cylindrical cloak. The two differential equations and two boundary conditions constitute a solvable system from which various cloak profiles can be designed. The proposed approach, which differs from the transformation optics approach, provides an intuitive and flexible design platform and allows considerable freedom to construct invisibility cloaks with a specific spatial distribution of material properties. The effects of the material properties and boundary conditions on cloak invisibility are analyzed in detail. Numerical simulations show that optimized cylindrical cloaks with finite material properties can be easily constructed.

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Transformation cloaking and radial approximations for flexural waves in elastic plates

Cited by 42 publications

References 40 publications

Platonic crystal with low-frequency locally-resonant spiral structures: wave trapping, transmission amplification, shielding and edge waves

Platonic crystal with low-frequency locally-resonant spiral structures: wave trapping, transmission amplification, shielding and edge waves

Omnidirectional flexural invisibility of multiple interacting voids in vibrating elastic plates

Manipulation of the propagation of out-of-plane shear waves

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