Transformation design of in-plane elastic cylindrical cloaks, concentrators and lenses

We analyse the elastic properties of a class of cylindrical cloaks deduced from linear geometric transforms x -> x ' in the framework of the Milton-Briane-Willis cloaking theory [New Journal of Physics 8, 248, 2006]. More precisely, we assume that the mapping between displacement fields u(x) -> u '(x ') is such that u '(x ') = A-tu(x), where A is either the transformation gradient Fij = 8x ' i/8xj or the second order identity tensor I. The nature of the cloaks under review can be three-fold: some of them are neutral for a source located a couple of wavelengths away; others lead to either a mirage effect or a field confinement when the source is located inside the concealment region or within their coated region (some act as elastic concentrators squeezing the wavelength of a pressure or shear polarized incident plane wave in their core); the last category of cloaks is classified as an elastic counterpart of electromagnetic perfect cylindrical lenses. The former two categories require either rank-4 elastic tensor and rank-2 density tensor and additional rank-3 and 2 positive definite tensors (A = F) or a rank-4 elasticity tensor and a scalar density (A = I) with spatially varying positive values. However, the latter example further requires that all rank-4, 3 and 2 tensors be negative definite (A = F) or that the elasticity tensor be negative definite (and non fully symmetric) as well as a negative scalar density (A = I). We provide some illustrative numerical examples with the Finite Element package Comsol Multiphysics when A is the identity.