  We present the realization of the topological valley-protected zero-order
antisymmetric (A0) or symmetric (S0) 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 A0,
S0, and SH0 Lamb waves at different domain walls in straight or Z-shape 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-shape 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 any more in that case.
