  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.
