Active cloaking of finite defects for flexural waves in elastic plates

We present a new method to create an active cloak for a rigid inclusion in a thin plate, and analyse flexural waves within such a plate governed by the Kirchhoff plate equation. We consider scattering of both a plane wave and a cylindrical wave by a single clamped inclusion of circular shape. In order to cloak the inclusion, we place control sources at small distances from the scatterer and choose their intensities to eliminate propagating orders of the scattered wave, thus reconstructing the respective incident wave. We then vary the number and position of the control sources to obtain the most effective configuration for cloaking the circular inclusion. Finally, we successfully cloak an arbitrarily shaped scatterer in a thin plate by deriving a semi-analytical, asymptotic algorithm.Comment: 19 pages, 14 figures, 1 tabl

Achieving control of in-plane elastic waves

We derive the elastic properties of a cylindrical cloak for in-plane coupled shear and pressure waves. The cloak is characterized by a rank 4 elasticity tensor with 16 spatially varying entries which are deduced from a geometric transform. Remarkably, the Navier equations retain their form under this transform, which is generally untrue [Milton et al., New J. Phys. 8, 248 (2006)]. We numerically check that clamped and freely vibrating obstacles located inside the neutral region are cloaked disrespectful of the frequency and the polarization of an incoming elastic wave.Comment: 9 pages, 4 figure

Eigenvalue problem in a solid with many inclusions: asymptotic analysis

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterised by a small parameter which is much larger compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.Comment: 55 pages, 5 figure

Wave Characterisation in a Dynamic Elastic Lattice: Lattice Flux and Circulation

A novel characterisation of dispersive waves in a vector elastic lattice is presented in the context of wave polarisation. This proves to be especially important in analysis of dynamic anisotropy and standing waves trapped within the lattice. The operators of lattice flux and lattice circulation provide the required quantitative description, especially in cases of intermediate and high frequency dynamic regimes. Dispersion diagrams are conventionally considered as the ultimate characteristics of dynamic properties of waves in periodic systems. Generally, a waveform in a lattice can be thought of as a combination of pressure-like and shear-like waves. However, a direct analogy with waves in the continuum is not always obvious. We show a coherent way to characterise lattice waveforms in terms of so-called lattice flux and lattice circulation. In the long wavelength limit, this leads to well-known interpretations of pressure and shear waves. For the cases when the wavelength is comparable with the size of the lattice cell, new features are revealed which involve special directions along which either lattice flux or lattice circulation is zero. The cases of high frequency and wavelength comparable to the size of the elementary cell are considered, including dynamic anisotropy and dynamic neutrality in structured solids

Trapped Modes and Steered Dirac Cones in Platonic Crystals

This paper discusses the properties of flexural waves obeying the biharmonic equation, propagating in a thin plate pinned at doubly-periodic sets of points. The emphases are on the properties of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates with a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that the Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.Comment: 21 pages, 12 figure

Periodically fighting shake, rattle and roll

How easy is it to suppress shake, rattle and roll in a long bridge or a skyscraper? Most practical structures are designed so that long wave resonance vibrations can be avoided. However, there are recent examples, such as the Millennium Bridge in London or the Volga Bridge in Volgograd, which show that unexpected external forces may result in large scale unwanted shake and rattle. Full scale alteration of a bridge (or a skyscraper) would not be considered as an acceptable option, unless the structure has collapsed. Can we fix this by examining a representative part of the structure only and making small lightweight changes? We will do it here and illustrate an idea linking the engineering analysis to elastic waveguides.Comment: 19 pages, 6 figure