Polarization-dependent and valley-protected Lamb waves in asymmetric pillared phononic crystals

We present the realization of the topological valley-protected zero-order antisymmetric (A(0)) or symmetric (S-0) and zero-order shear-horizontal (SH0) Lamb waves at different domain walls based on topologically distinct asymmetric double-sided pillared phononic crystals. The elastic periodic structures have either the triangular or the honeycomb symmetry and give rise to a double-negative branch in the dispersion curves. By artificially folding the doubly negative branch, a degenerate Dirac cone is achieved. Different polarization-dependent propagation along the same primary direction along the constituent branches are presented. Moreover, divergent polarization-dependent phenomena along different primary directions along a given branch are also reported. By imposing two large space-inversion symmetry (SIS) breaking perturbations the topological phase transition is obtained. We show that the Berry curvature becomes strongly anisotropic when the wave vector gets away from the valleys. Further, we demonstrate the unidirectional transport of A(0), S-0, and SH0 Lamb waves at different domain walls in straight or Z-shaped wave guides. In the large SIS breaking case, we show negligible reflection at the zigzag outlet of the straight wave guide and occurrence of weak inter-valley scattering at the bending corners of the Z-shaped wave guide. For a larger strength of SIS breaking, the edge states are gapped and strong reflection at the zigzag outlet and bending corners is observed. The topological protection cannot be guaranteed anymore in that case.