EUROMECH Colloquium 540

The principle underlying cloaking theory is the transformation method whereby the material properties of the cloak are defined by a (singular) spatial transformation. For elastodynamics, Milton et al. [1] concluded that the transformed materials are described by the Willis model, involving coupling between stress and velocity, in addition to anisotropic inertia. For a restricted transformation, Brun et al. [2] found transformed material properties with isotropic inertia and elastic behavior of Cosserat type, i.e. with properties that are the same as those of “standard” linear elasticity except that the moduli do not satisfy the minor symmetry, i.e. C∗ jikl 6= C∗ ijkl. The transformed elastodynamic constitutive parameters may be characterized through their dependence on (i) the transformation (mapping function) and (ii) on the relation between the displacement fields in the two descriptions, represented by matrices: F, the deformation gradient matrix, and A, respectively. It was shown [3] that requiring stress to be symmetric implies A = F and that the material must be of Willis form, as in [1]. Setting A = I, on the other hand, results in Cosserat materials with non-symmetric stress but isotropic density, as found by Brun et al. [2]. In this paper we consider a class of materials displaying non-symmetric stress of the type necessary to achieve elastodynamic cloaking by taking advantage of the similarities between transformation elasticity and small-on-large motion in the presence of finite pre-strain [4]. Such an approach has already been shown to be successful; by using an incompressible neo-Hookean material with a radially symmetric cylindrical pre-strain, Parnell [5] showed that the resulting small-on-large equations are identically those required for cloaking of the horizontally polarized shear (SH) wave motion. Here we consider the more general elastic transformation problem, including but not limited to SH motion.