The topological interface state governed by topological phononic crystals (PnC) can
potentially host one-way, backscattering free nontrivial edge modes, immune to
defects and sharp edges. We study here 1D topological phononic crystals with
interface modes/states generated by an exchange of wave mode polarization and
geometric phases, using the spectral element method with Timoshenko beam model
for flexural wave propagation. The constitutive relations for the longitudinal wave,
and modeling and formulation are derived for theoretical band structure and
frequency response studies. The analysis is validated by finite element numerical
simulations. The geometric phases of the Bloch bands are determined by numerical
Zak phase analysis. As the geometric properties of the PnC vary, a band transition
resulting from an exchange in wave mode polarization is observed and the symmetry
characteristics of the Bloch bands are determined. The geometric phases provide
useful information about the interface mode that is generated when the
mode transition frequency is common between the bandgaps of topological
PnC. We further conduct theoretical and numerical studies on the presence
of interface state and excellent agreement observed between both models
is reported. The theoretical details of the topological PnC with protected
interface mode can be helpful for better understating of research in phononic
crystals.
