Static elastic cloaking, low-frequency elastic wave transparency and neutral inclusions.

Andrew N Norris,William J Parnell

doi:10.1098/rspa.2019.0725

Abstract

New connections between static elastic cloaking, low-frequency elastic wave scattering and neutral inclusions (NIs) are established in the context of two-dimensional elasticity. A cylindrical core surrounded by a cylindrical shell is embedded in a uniform elastic matrix. Given the core and matrix properties, we answer the questions of how to select the shell material such that (i) it acts as a static elastic cloak, and (ii) it eliminates low-frequency scattering of incident elastic waves. It is shown that static cloaking (i) requires an anisotropic shell, whereas scattering reduction (ii) can be satisfied more simply with isotropic materials. Implicit solutions for the shell material are obtained by considering the core–shell composite cylinder as a neutral elastic inclusion. Two types of NI are distinguished, weak and strong with the former equivalent to low-frequency transparency and the classical Christensen and Lo generalized self-consistent result for in-plane shear from 1979. Our introduction of the strong NI is an important extension of this result in that we show that standard anisotropic shells can act as perfect static cloaks, contrasting previous work that has employed ‘unphysical’ materials. The relationships between low-frequency transparency, static cloaking and NIs provide the material designer with options for achieving elastic cloaking in the quasi-static limit.

Highlights

The ability to cloak a region of space so that an incident field or applied loading does not see or feel the presence of an object is of great interest in science and engineering
Static cloaking is closely related with the concept of a neutral inclusion (NI), which is a region of inhomogeneity in an otherwise uniform solid that does not disturb an applied exterior field
We provide examples of composite cylinders comprising isotropic cores and uniform anisotropic shells that are strong NIs and illustrate two of these cases graphically showing the difference between the weak and strong NI in the process

Summary

Introduction

The ability to cloak a region of space so that an incident field or applied loading does not see or feel the presence of an object is of great interest in science and engineering. By inserting the composite cylinder (core plus coating) in the background medium and requiring that it act as an NI, the background properties provide a self-consistent estimate of the effective material properties, i.e. the matrix properties. This approach partitions into two sub-problems, first for in-plane hydrostatic loading which ensures a condition can be determined for the effective bulk modulus. The problem considered is as follows: given a host matrix and a cylindrical core, determine the shell properties such that the core+shell (composite assemblage) acts as either a strong or a weak NI, to planar deformation of hydrostatic and shear loading (figure 1).

Static cloak using an approximate model

Quasi-static cloaking problem set-up

Implementation of the theory: weak versus strong neutral inclusions

Findings

Conclusion