On near-cloaking for linear elasticity

Abstract

We make precise some results on the cloaking of displacement fields in linear elasticity. In the spirit of transformation media theory, the transformed governing equations in Cosserat and Willis frameworks are shown to be equivalent to certain high contrast small defect problems for the usual Navier equations. We discuss near-cloaking for elasticity systems via a regularized transform and perform numerical experiments to illustrate our near-cloaking results. We also study the sharpness of the estimates from [H. Ammari, H. Kang, K. Kim and H. Lee, J. Diff. Eq. 254, 4446-4464 (2013)], wherein the convergence of the solutions to the transmission problems is investigated, when the Lamé parameters in the inclusion tend to extreme values. Both soft and hard inclusion limits are studied and we also touch upon the finite frequency case. Finally, we propose an approximate isotropic cloak algorithm for a symmetrized Cosserat cloak.

7 figures, 7 tables; Note that the earlier version of this preprint was titled 'Some results in near-cloaking for elasticity systems'. This new version of the manuscript has also seen some major upgrade. We have added a new section on 'Cloaking parameters and isotropic approximation'. In there, we propose an approximate isotropic cloak algorithm for a symmetrized Cosserat cloak