Discrete transformation elasticity: An approach to design lattice-based polar metamaterials

The transformation method is a powerful tool providing the constitutive parameters necessary for arbitrary geometric transformations of solution fields. These constitutive parameters, in elasticity, describe a constitutive law that, unlike conventional Hooke's law, is polar and chiral and that no known solids exhibit. This raises the question of whether polar and chiral elastic solids can be designed from the bottom up as architected lattice-based materials; this design task is a major challenge in the field of transformation elasticity. The present study aims to provide a theoretically justified design methodology based on a discrete transformation method. The key idea is to let the gradient of the geometric transformation operate not only on the elastic properties but on the underlying lattice-based architecture of the solids. As an outstanding application, we leverage the proposed design paradigm to construct a polar lattice metamaterial subsequently used for elastic carpet cloaking purposes. Numerical simulations are carried to show excellent cloaking performance under different static and dynamic mechanical loads and thus demonstrate the validity of the proposed designs. The approach presented herein could promote and accelerate new designs of lattice topologies for transformation elasticity in particular and can be extended to realize other emergent elastic properties and to unlock peculiar field-warping functions other than cloaking in static and dynamic contexts.