Symmetry-induced collapse of elastic polarization in a strain–energy Hilbert space

Abstract Elastic waves intrinsically support multiple symmetry-distinct deformation channels, rendering polarization control fundamentally more complex than in scalar or purely transverse wave systems. While polarization in electromagnetic waves is uniquely defined by the orientation of a transverse field vector, elastic polarization arises from the coexistence of volumetric and deviatoric strain representations that may occupy the same frequency and wavevector. Here we reformulate elastic polarization as an admissibility problem in a strain–energy Hilbert space endowed with a metric defined by the elastic stiffness tensor. We show that symmetry and Bloch periodicity can collapse this Hilbert space, enforcing a unique admissible strain representation within finite frequency windows of a three-dimensional elastic metamaterial. Using a meta-atom that engineers the hierarchy of strain–energy metrics through directional bending inertia, we demonstrate complete non-admissibility of competing elastic polarizations along all principal directions of the Brillouin zone. We further show that the identity of the admissible elastic polarization is governed by the preservation or breakdown of this strain–energy hierarchy, independent of absolute resonance frequency. These results establish elastic polarization as a symmetry-governed state-space phenomenon and introduce Hilbert-space collapse as a general mechanism for deterministic elastic wave control.