Topological mechanics of edge waves in Kagome lattices

Topological insulators are new phases of matter whose properties are derived from a number of qualitative yet robust topological invariants rather than specific geometric features or constitutive parameters. Here, Kagome lattices are classified based on a topological invariant directly related to the handedness of a couple of elliptically polarized stationary eigenmodes in the context of what is known as the "quantum valley Hall effect" in physics literature. An interface separating two topologically distinct lattices, i.e., two lattices with different topological invariants, is then proven to host two topological Stoneley waves whose frequencies, shapes and decay and propagation velocities are quantified. Conversely, an interface separating two topologically equivalent lattices will host no Stoneley waves. Analysis is based on an asymptotic model derived through a modified high-frequency homogenization procedure. This case study constitutes the first implementation of the quantum valley Hall effect in in-plane elasticity. A preliminary discussion of 1D lattices is included to provide relevant background on topological effects in a simple analytical framework.