Nearly cloaking for the elasticity system with residual stress

TL;DR

This paper extends nearly cloaking techniques to anisotropic elastic systems with residual stress, overcoming theoretical challenges by deriving new elliptic estimates without layer potential theory.

Contribution

It introduces a novel approach to nearly cloaking for residual stress elasticity systems, which are more natural and complex than isotropic models, using elliptic estimates instead of layer potential theory.

Findings

Successfully extended cloaking to residual stress systems

Developed elliptic estimates for anisotropic elasticity without layer potentials

Demonstrated invariance under coordinate transformations

Abstract

The nearly cloaking via transformation optics approach for the isotropic elastic wave fields is considered. This work extends the study of the nearly cloaking scheme to the elasticity system with residual stress in \mathbb{R}^{N} for N=2,3, which is an anisotropic elasticity system. It is worth to mention that there are no minor symmetric properties for the elastic tensor with residual stress and this system is invariant under a coordinate transformation. Therefore, we think the elasticity system residual stress model is more natural than the isotropic elasticity system in designing the elastic cloaking medium in the physical sense. In addition, the main difficulty of treating this problem lies in the fact that there are no layer potential theory for the residual stress system. Instead, we will derive suitable elliptic estimates for the elasticity system residual stress by comparing with the Lam\'e system to achieve desired results.