Hyperelastic cloaking theory: Transformation elasticity with pre-stressed solids

TL;DR

This paper develops a hyperelastic cloaking theory by linking transformation elasticity with pre-stressed solids, enabling wave cloaking through finite pre-strain and specific strain energy functions.

Contribution

It establishes a formal equivalence between transformation elasticity and incremental motion on pre-stressed hyperelastic solids, introducing a method to achieve cloaking via finite strains.

Findings

Transformation elasticity parameters correspond to small-on-large theory.

Cloaking can be achieved with specific hyperelastic strain energy functions.

Numerical examples demonstrate cloaking of wave motion.

Abstract

Transformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting the similarity between transformation elasticity and the theory of incremental motion superimposed on finite pre-strain it is shown that the constitutive parameters of transformation elasticity correspond to the density and moduli of small-on-large theory. The formal equivalence indicates that transformation elasticity can be achieved by selecting a particular finite (hyperelastic) strain energy function, which for isotropic elasticity is semilinear strain energy. The associated elastic transformation is restricted by the requirement of statically equilibrated pre-stress. This constraint can be cast as $\tr {\mathbf F} =$ constant, where $\mathbf{F}$ is the deformation gradient, subject to symmetry constraints, and its consequences are explored both analytically and through numerical examples of cloaking of anti-plane and in-plane wave motion.