Discrete Transformation Elasticity: An Approach to Design Lattice-based Polar Metamaterials

The transformation method is a powerful tool for providing the constitutive parameters of the transformed material in the new coordinates. In transformation elasticity, a general curvilinear change of coordinates transforms conventional Hooke's law into a different constitutive law in which the transformed material is not only anisotropic but also polar and chiral and no known elastic solid satisfies. However, this state-of-the-art description provides no insight as to what the underlying microstructure of this transformed material could be, the design of which is a major challenge in this field. The study aims to theoretically justify the fundamental need for the polar material by critically revisiting the discrete transformation method. The key idea is to let transformation gradient operate not only on the elastic properties but on the underlying architectures of the mechanical lattice. As an outstanding application, we leverage the proposed design paradigm to physically construct a polar lattice metamaterial for the observation of elastic carpet cloaking. Numerical simulations are then implemented to show excellent cloaking performance under different static and dynamic mechanical loads. The approach presented herein could promote and accelerate new designs of lattice topologies for transformation elasticity in particular and is able to be extended for realizing other emerging elastic properties and unlocking peculiar functions including statics and dynamics in general.