Higher-Order Topological Phase Transitions in Continuous Hyperelastic Manifolds: From Surface Wrinkles to Zero-Energy Corner States

Abstract

Higher-order topological insulators (HOTIs) have revolutionized our understanding of wave localization, extending the bulk-boundary correspondence to lower-dimensional hinges and corners. Thus far, the realization of mechanical HOTIs has relied exclusively on discretely engineered metamaterials or periodic phononic lattices. Here, we report a fundamental paradigm shift by demonstrating that continuous, homogeneous hyperelastic manifolds undergoing finite multiaxial deformations naturally harbor intrinsic higher-order topological phases. By extending the generalized Stroh-Lie impedance formalism into a fully coupled 3D finite-strain framework, we map the highly nonlinear orthotropic geometric frustration onto a four-band effective Dirac Hamiltonian spanned by Clifford $\Gamma$-matrices. We reveal that macroscopic orthogonal stretches act precisely as competing Dirac mass terms, driving the continuous spatial transitions of topological domain walls and triggering a breakdown of $C_{4v}$ spatial symmetry. Remarkably, we analytically prove that beyond classical 2D surface wrinkling (1st-order topology), concurrent multiaxial extreme compression unconditionally triggers the emergence of 1D hinge states (2nd-order) and completely localized 0D zero-energy corner states (3rd-order). We further extend this static bifurcation framework into the elastodynamic regime, proving the existence of mid-gap localized vibrational modes. The theoretically derived topological phase diagram, nested Wilson loops, and fractional corner charges are comprehensively verified. Finally, we propose a concrete experimental realization using electro-active dielectric elastomers, enabling the dynamic programming of 0D topological singularities.