  We report a new type of phononic crystals with topologically non-trivial
bandgaps for both longitudinal and transverse polarizations, resulting in
protected one-way elastic edge waves. In our design, gyroscopic inertial
effects are used to break the time-reversal symmetry and realize the phononic
analogue of the electronic quantum Hall effect. We investigate the response of
both hexagonal and square gyroscopic lattices and observe bulk Chern number of
1 and 2, indicating that these structures support single and multi-mode edge
elastic waves immune to back-scattering. These robust one-way phononic
waveguides could potentially lead to the design of a novel class of surface
wave devices that are widely used in electronics, telecommunication and
acoustic imaging.
