Predictive beam-lattice reduction for higher-order topological modes in a 2D SSH phononic crystal

Abstract

We develop a mechanically faithful reduced model for a two-dimensional topological phononic crystal composed of rigid square masses connected by slender elastic ligaments. Exploiting Euler Bernoulli beam theory, we derive a Hermitian 12 degree of freedom dynamical matrix that retains in-plane translations, rotations, and ligament eccentricity. This reduction captures effects that are absent from scalar mass spring SSH models while remaining computationally much more tractable and more easily interpretable than full finite element simulations. Dimerizing the ligament widths produces a mechanical 2D SSH lattice with a full band gap and a quantized bulk polarization. The sign of the dimerization controls the transition from trivial to non trivial phases, while ligament eccentricity provides an additional purely geometric mechanism for changing the topology. Ribbon and finite cell calculations predict in gap edge and corner modes, quantified by localization measures and confirmed by finite element simulations. Measurements on 3D printed samples show an evanescent response in the trivial structure and enhanced boundary/corner response in the non trivial structure within the predicted gap. The results provide a validated route for designing topological elastic metamaterials using a continuum informed discrete model rather than either idealized mass spring networks or brute force numerical optimization.