Controlling solid elastic waves with spherical cloaks

We propose a cloak for coupled shear and pressure waves in solids. Its elastic properties are deduced from a geometric transform that retains the form of Navier equations. The spherical shell is made of an anisotropic and heterogeneous medium described by an elasticity tensor C' (without the minor symmetries) which has 21 non-zero spatially varying coefficients in spherical coordinates. Although some entries of C, e.g. some with a radial subscript, and the density (a scalar radial function) vanish on the inner boundary of the cloak, this metamaterial exhibits less singularities than its cylindrical counterpart studied in [M. Brun, S. Guenneau, A.B. Movchan, Appl. Phys. Lett. 94, 061903 (2009).] In the latter work, C' suffered some infinite entries, unlike in our case. Finite element computations confirm that elastic waves are smoothly detoured around a spherical void without reflection.Comment: Version 3: minor typos corrected. Figures captions improved. 5 figures. Key words: 3D elastic cloaking, seismic metamaterials. This paper is the cover of the 14 July 2014 issue of Applied Physics Letter

Cloaking via change of variables in elastic impedance tomography

We discuss the concept of cloaking for elastic impedance tomography, in which, we seek information on the elasticity tensor of an elastic medium from the knowledge of measurements on its boundary. We derive some theoretical results illustrated by some numerical simulations.Comment: latex, 2 figures, 11 pages, submitte

Classical Yang-Baxter Equation and Left Invariant Affine Geometry on Lie Groups

Let G be a Lie group with Lie algebra \Cal G: = T_\epsilon G and T^*G = \Cal G^* \rtimes G its cotangent bundle considered as a Lie group, where G acts on \Cal G^* via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions r\in \wedge^2 \Cal G of the Classical Yang-Baxter Equation in G, and the set of connected Lie subgroups of $T^*G$ which carry a left invariant affine structure and whose Lie algebras are lagrangian graphs in \Cal G \oplus \Cal G^*. An invertible solution r endows G with a left invariant symplectic structure and hence a left invariant affine structure. In this case we prove that the Poisson Lie tensor $\pi := r^+ - r^-$ is polynomial of degree at most 2 and the double Lie groups of $(G,\pi)$ also carry a canonical left invariant affine structure. In the general case of (non necessarly invertible) solutions r, we supply a necessary and suffisant condition to the geodesic completness of the associated affine structureComment: 13 pages, late

Lattices in contact Lie groups and 5-dimensional contact solvmanifolds

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice subgroup. We re-examine Alexander's criteria for existence of lattices on solvable Lie groups and apply them, along with some other well known tools, and use these results to prove that, in dimension 5, there are exactly seven connected and simply connected contact Lie groups with uniform lattices, all of which are solvable. Issues of symplectic boundaries are explored, as well. It is also shown that the special affine group has no uniform lattice.Comment: V3: 19 pages. Latex. Former Section 5 from V1 and V2 has now been removed. We thank Dr Chris Wendl and Prof. Patrick Massot for constructive remarks that have led to some reconsiderations on former Section 5 of V2. Last version appeared at Kodai Mathematical Journa

On properties of principal elements of Frobenius Lie algebras

We investigate the properties of principal elements of Frobenius Lie algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that any Lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on simple Lie algebras, applied to the particular case of sl(m, K) alone, paves the way to the complete classification of Frobenius and more generally quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the property that every derivation is an inner derivation, then every principal element is semisimple, at least for K=C. As an important case, we prove that in the Lie algebra of the group of affine motions of the Euclidean space of finite dimension, every derivation is inner. We also bring a class of examples of Frobenius Lie algebras, that hence are subalgebras of sl(m, K), but yet have nonsemisimple principal elements as well as some with semisimple principal elements having nonrational eigenvalues, where K=R or C.Comment: Latex, 16 pages. The last version appeared at Journal of Lie Theory. Keywords and phrases: Frobenius Lie algebra, affine Lie algebra, Left symmetric Lie algebra, affine motion, symplectic Lie algebra, seaweed Lie algebra, symplectic Lie group, invariant symplectic structure, invariant affine structur

Elastodynamic cloaking and field enhancement for soft spheres

In this paper, we bring to the awareness of the scientific community and civil engineers, an important fact: the possible lack of wave protection of transformational elastic cloaks. To do so, we propose spherical cloaks described by a non-singular asymmetric elasticity tensor depending upon a small parameter $\eta,$ that defines the softness of a region one would like to conceal from elastodynamic waves. By varying $\eta$, we generate a class of soft spheres dressed by elastodynamic cloaks, which are shown to considerably reduce the soft spheres' scattering. Importantly, such cloaks also provide some wave protection except for a countable set of frequencies, for which some large elastic field enhancement (resonance peaks) can be observed within the cloaked soft spheres, hence entailing a possible lack of wave protection. We further present an investigation of trapped modes in elasticity via which we supply a good approximation of such Mie-type resonances by some transcendental equation. Next, after a detailed presentation of spherical elastodynamic PML of Cosserat type, we introduce a novel generation of cloaks, the mixed cloaks, as a solution to the lack of wave protection in elastodynamic cloaking. Indeed, mixed cloaks achieve both invisibility cloaking and protection throughout a large range of frequencies. We think, mixed cloaks will soon be generalized to other areas of physics and engineering and will in particular foster studies in experiments.Comment: V2: major changes. More details on the study of trapped modes in elasticity. Mixed cloaks introduced. Latex files, 27 pages, 14 figures. The last version will appear at Journal of Physics D: Applied Physics. Pacs:41.20.Jb,42.25.Bs,42.70.Qs,43.20.Bi,43.25.Gf. arXiv admin note: text overlap with arXiv:1403.184

Steering in-plane shear waves with inertial resonators in platonic crystals

Numerical simulations shed light on control of shear elastic wave propagation in plates structured with inertial resonators. The structural element is composed of a heavy core connected to the main freestanding plate through tiny ligaments. It is shown that such a configuration exhibits a complete band gap in the low frequency regime. As a byproduct, we further describe the asymmetric twisting vibration of a single scatterer via modal analysis, dispersion and transmission loss. This might pave the way to new functionalities such as focusing and self-collimation in elastic plates