  Topological phononic crystals, alike their electronic counterparts, are
characterized by a bulk-edge correspondence where the interior of a material
dictates the existence of stable surface or boundary modes. In the mechanical
setup, such surface modes can be used for various applications such as
wave-guiding, vibration isolation, or the design of static properties such as
stable floppy modes where parts of a system move freely. Here, we provide a
classification scheme of topological phonons based on local symmetries. We
import and adapt the classification of non-interacting electron systems and
embed it into the mechanical setup. Moreover, we provide an extensive set of
examples that illustrate our scheme and can be used to generate new models in
unexplored symmetry classes. Our works unifies the vast recent literature on
topological phonons and paves the way to future applications of topological
surface modes in mechanical metamaterials.
