Cloaking In-Plane Elastic Waves with Swiss Rolls

We propose a design of cylindrical cloak for coupled in-plane shear waves consisting of concentric layers of sub-wavelength resonant stress-free inclusions shaped as Swiss rolls. The scaling factor between inclusions’ sizes is according to Pendry’s transform. Unlike the hitherto known situations, the present geometric transform starts from a Willis medium and further assumes that displacement fields u in original medium and u ′ in transformed medium remain unaffected ( u ′ = u ). This breaks the minor symmetries of the rank-4 and rank-3 tensors in the Willis equation that describe the transformed effective medium. We achieve some cloaking for a shear polarized source at specific, resonant sub-wavelength, frequencies, when it is located in close proximity to a clamped obstacle surrounded by the structured cloak. The structured medium approximating the effective medium allows for strong Willis coupling, notwithstanding potential chiral elastic effects, and thus mitigates roles of Willis and Cosserat media in the achieved elastodynamic cloaking.

Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

A Lie groupoid, called second-order non-holonomic material Lie groupoid, is associated in a natural way to any Cosserat medium. This groupoid is used to give a new definition of homogeneity which does not depend on a material archetype. The corresponding Lie algebroid, called second-order non-holonomic material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously obtained ones in terms of non-holonomic second-order [Formula: see text]-structures.

Higher Order Frames and Material Symmetries

The paper is focused on mathematical approach to materials with microstructure, the special case of them are Cosserat media. Frame bundles are described with respect to their role in continuum mechanics and the structure jet groups. General (r-th order) holonomic, nonholonomic and semiholonomic microstructure configuration are introduces and the algebraic approach using Weil algebras is presented, too.