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

TL;DR

This paper analyzes the elastic properties of cylindrical cloaks, concentrators, and lenses based on geometric transforms within the Milton-Briane-Willis theory, revealing their potential to manipulate elastic waves and their specific material requirements.

Contribution

It introduces a comprehensive analysis of elastic cylindrical cloaks, concentrators, and lenses using geometric transforms, highlighting their different wave manipulation capabilities and material property requirements.

Findings

Some cloaks are neutral for distant sources.

Others create mirage effects or confine fields.

Certain cloaks act as elastic lenses with negative material properties.

Abstract

We analyse the elastic properties of a class of cylindrical cloaks deduced from linear geometric transforms ${\bf x} \to {\bf x}'$ 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 ${\bf u}({\bf x}) \to {\bf u}'({\bf x}')$ is such that ${\bf u}'({\bf x}') = {\bf A}^{-t}{\bf u}({\bf x})$, where ${\bf A}$ is either the transformation gradient $F_{ij} = \partial x'_i/ \partial x_j$ or the second order identity tensor ${\bf I}$. 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; other 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); a 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 $({\bf A} = {\bf F})$ or a rank 4 elasticity tensor and a scalar density $({\bf A} = {\bf I})$ with spatially varying positive values. However, the latter example further requires that all rank-4, 3 and 2 tensors be negative definite $({\bf A} = {\bf F})$ or that the elasticity tensor be negative definite (and non fully symmetric) as well as a negative scalar density $({\bf A} = {\bf I})$. We provide some illustrative numerical examples with the Finite Element package Comsol Multiphysics when ${\bf A}$ is the identity.