Article

Cloaking strategy for Love waves

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Abstract

Love waves are antiplane elastic waves which propagate along the surface of a heterogeneous medium. Under time-harmonic regime, they are governed by a scalar equation of the Helmholtz type. We exploit the invariance of this governing equation under an in-plane arbitrary coordinate transformation to design broadband cloaks for surface defects. In particular, we apply transformation elastodynamics to determine the anisotropic, position dependent, mechanical properties of ideal cloaks able to hide triangular and parabolic-shaped defects. Dispersion analysis and time-harmonic numerical simulations are employed to validate the proposed strategy. Next, we utilize layered monoclinic materials, with homogenized properties matching those of the ideal cloaks, to design feasible triangular-shaped cloaks. The performance of the layered cloaks is validated via parametric analysis of the dispersion curves, which converge to those of the ideal cloak when the unit cell-wavelength ratio vanishes. Finally, time-harmonic numerical simulations confirm a significant reduction of the defect-generated scattered fields by the layered cloaks.

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... In this study we recalled a brief theoretical background and developed a strategy for 3-D elastic cloaking from surface waves, assuming cylindrical symmetry. This idea is an expansion of [4] where only the case of Love waves was considered. We believe that this work can support future extensions, including the realization of a fully feasible and functional elastic cloaking device. ...
... In the context of elastodynamics, materials dressed with space, time, and space-time modulations are conveniently molded to manipulate wave motion at will. Examples have shown different forms of waveguiding that rely on defect mechanisms [1], topological insulators [2][3][4][5][6][7][8][9][10], metagradings [11][12][13][14][15][16][17], transformation optics and acoustics [18][19][20][21][22][23][24][25][26][27][28], which are passive strategies effectively employed to accomplish relevant technological behaviors, such as communication, focusing, and invisibility. ...
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The transformation theory of optics and acoustics is developed for the equations of linear anisotropic elasticity. The transformed equations correspond to non-unique material properties that can be varied for a given transformation by selection of the matrix relating displacements in the two descriptions. This gauge matrix can be chosen to make the transformed density isotropic for any transformation although the stress in the transformed material is not generally symmetric. Symmetric stress is obtained only if the gauge matrix is identical to the transformation matrix, in agreement with Milton et al. (2006). The elastic transformation theory is applied to the case of cylindrical anisotropy. The equations of motion for the transformed material with isotropic density are expressed in Stroh format, suitable for modeling cylindrical elastic cloaking. It is shown that there is a preferred approximate material with symmetric stress that could be a useful candidate for making cylindrical elastic cloaking devices.
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Control of waves with metamaterials is of great topical interest, and is fueled by rapid progress in broadband acoustic and electromagnetic cloaks. We propose a design for a cloak to control bending waves propagating in isotropic heterogeneous thin plates. This is achieved through homogenization of a multilayered concentric coating filled with piecewise constant isotropic elastic material. Significantly, our cloak displays no phase shift for both backward and forward scattering. To foster experimental efforts, we provide a simplified design of the cloak which is shown to work in a more than two-octave frequency range (30 Hz to 150 Hz) when it consists of 10 layers using only 6 different materials overall. This metamaterial should be easy to manufacture, with potential applications ranging from car industry to anti-earthquake passive systems for smart buildings, depending upon the plate dimensions and wavelengths.
Article
We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.
Article
An acoustic cloak envelopes an object so that sound incident from all directions passes through and around the cloak as though the object were not present. A theory of acoustic cloaking is developed using the transformation or change-of-variables method for mapping the cloaked region to a point with vanishing scattering strength. We show that the acoustical parameters in the cloak must be anisotropic: either the mass density or the mechanical stiffness or both. If the stiffness is isotropic, corresponding to a fluid with a single bulk modulus, then the inertial density must be infinite at the inner surface of the cloak. This requires an infinitely massive cloak. We show that perfect cloaking can be achieved with finite mass through the use of anisotropic stiffness. The generic class of anisotropic material required is known as a pentamode material. If the transformation deformation gradient is symmetric then the pentamode material parameters are explicit, otherwise its properties depend on a stress like tensor which satisfies a static equilibrium equation. For a given transformation mapping the material composition of the cloak is not uniquely defined, but the phase and wave speeds of the pseudo-acoustic waves in the cloak are unique. Examples are given from 2D and 3D.
On form invariance of the kirchhoff-love plate equation
  • L Pomot
  • S Bourgeois
  • C Payan
  • M Remillieux
  • S Guenneau
L. Pomot, S. Bourgeois, C. Payan, M. Remillieux, S. Guenneau, On form invariance of the kirchhoff-love plate equation, 2019, arXiv preprint arXiv: 1901.00067.
The Theory of Composites
  • G W Milton
G.W. Milton, The Theory of Composites, 2002.