  We realize an elastic second-order topological insulator hosting both
one-dimensional gapped edge states and zero-dimensional in-gap corner modes in
the double-sided pillared phononic crystal plates with square lattice. Changing
the width of two neighbor pillars breaks the inversion symmetry and induces the
band inversion to emulate the quantum spin Hall effect where the gapless edge
states are obtained. Further breaking the space-symmetry at interface, the
gapless edge states are gapped and inducing the edge topological transitions
and then giving rise to the zero-dimensional in-gap corner modes. Our work
offers a novel way for elastic wave trapping and robustly guiding.
