  Elastic material with its elastic tensor losing minor symmetry is considered
impossible without introducing artificially body torque. Here we demonstrate
the feasibility of such material by introducing rotational resonance, the
amplified rotational inertia of the microstructure during dynamical loading
breaks naturally the shear stress symmetry, without resorting to external body
torque or any other active means. This concept is illustrated through a
realistic mass-spring model together with analytical homogenization technique
and band structure analysis. It is also proven that this metamaterial model can
be deliberately tuned to meet the material requirement defined by
transformation method for full control of elastic wave, and the relation
bridging the microstructure and the desired wave functionality is explicitly
given. Application of this asymmetric metamaterial to design elastic wave cloak
is demonstrated and validated by numerical simulation. The study paves the way
for material design used to construct the transformation media for controlling
elastic wave and related devices.
