Hyperelastic cloaking theory: Transformation elasticity with prestressed solids

Overview & Synopsis

Interest in cloaking theory (rendering objects near-invisible to incident waves) and its practical realization has grown significantly since the early theoretical work in 2006 of Leonhardt and the Pendry group in optics and electromagnetism respectively. Methods have largely been based on the idea of coordinate transformations, which motivate the design of cloaking metamaterials. These materials are able to guide waves around a specific region of space. Research has subsequently focused on the possibility of cloaking in the contexts of acoustics, surface waves in fluids, heat transfer, fluid flow and linear elastodynamics. It was shown by Milton and co-workers that elastodynamic cloaking is made difficult due to the lack of invariance of Navier's equations under general coordinate transformations. Restricted mappings do preserve the form of equations but the transformed elastic modulus tensor does not possess the minor symmetries. We shall show that it is theoretically possible to construct elastodynamic cloaks by pre-stressing hyperelastic solids. We discuss an initial simple case (antiplane waves) [Parnell, W.J. "Nonlinear pre-stress for cloaking from antiplane elastic waves", Proc Roy Soc A, online version: doi: 10.1098/rspa.2011.0477] and describe various generalizations including finite antiplane cloaks and more general elastodynamic problems.