Isotropic polar solids for conformal transformation elasticity and cloaking

TLDR: In this paper, a constitutive theory of isotropic linear elastic solids where Cauchy's stress tensor is not necessarily symmetric thus earning them the attribute "polar" was proposed.

Abstract: The paper initiates a constitutive theory of isotropic linear elastic solids where Cauchy’s stress tensor is not necessarily symmetric thus earning them the attribute “polar”. Mainly, expressions for the fourth-order elasticity tensors of isotropic polar solids in two and three space dimensions are derived. It is found that the constitutive tensors feature, in addition to the usual bulk and shear moduli, one extra independent elastic constant in 3D, and two extra constants in 2D. In the latter case, the new constants quantify to which degree stress and mirror symmetries are broken. Indeed, it turns out that in 2D, stress asymmetry enables isotropic yet chiral behaviors and the interplay between chirality and polarity is subsequently investigated. To motivate the theory, it is shown that 2D isotropic polar solids naturally arise in the application of the transformation method when the background medium is isotropic and when the underlying spatial transformation is conformal (i.e., angle-preserving). Accordingly, isotropic polar solids, in conjunction with conformal transformations, can simplify the design of invisibility cloaks and other wave-steering devices by circumventing the need for anisotropic behaviors. As a demonstration, a conformal carpet cloak is designed out of graded hexachiral lattices and numerically tested. The cloaking performance is shown to be satisfactory over a range of pressure-shear coupled dynamic loadings.