Elastic Invisibility in Three Dimensions: Neutral Inclusions and Low Frequency Transparency

Summary We address the topic of elastostatic invisibility, or cloaking, in three dimensions, where an inclusion is coated with some medium to ensure that it is invisible to imposed forces in the far-field. Typically this effect requires the coating to be an inhomogeneous and anisotropic material, with examples generated via transformation elastostatics, and often with unphysical properties. Such complex designs have limited applicability to the manufacture of real-world materials. In materials science and engineering, the neutral inclusion concept offers the more realistic aim of ensuring that coated inclusions are invisible to specified loading states. Here we demonstrate that it is possible to make a spherical inclusion invisible to any linear combination of hydrostatic and shear deformation by surrounding it with an appropriate spherically transversely isotropic coating. It is additionally shown that two isotropic layers can have an equivalent neutralising effect. Finally, we demonstrate the links with low frequency transparency by considering the dynamic analogue to the neutral inclusion setup at low frequency. In particular, it is shown that low frequency transparency, achieved by eliminating the leading order scattering coefficients, may fail to neutralise the leading order perturbed field local to the obstacle.