  The study of topological states in electronic structures, which allows robust
transport properties against impurities and defects, has been recently extended
to the realm of elasticity. This work shows that nontrivial topological
flexural edge states located on the free boundary of the elastic graphene-like
metamaterial can be realized without breaking the time reversal, mirror, or
inversion symmetry of the system. Numerical calculations and experimental
studies demonstrate the robust transport of flexural waves along the boundaries
of the designed structure. The topological edge states on the free boundary are
not limited by the size of the finite structure, which can reduce the scale of
the topological state system. In addition, unlike the edge states localized on
the free boundary in graphene where the group velocity is zero, the edge states
on the elastic metamaterial plate have propagation states with non-zero group
velocity. There is a frequency range for the edge states, and we introduce the
concept of Shannon entropy for elastic waves and use it to assess the frequency
range of the edge states in graphene-like elastic metamaterials. This work
represents a relevant advance in the study of elastic wave topological states,
providing a theoretical basis for engineering applications such as vibration
reduction and vibration isolation of mechanical structures.
