Elastodynamics of two-dimensional odd mechanical media: Theory, homogenization and wave characterization

Non-Hermitian systems have attracted growing attention in mechanics due to their unique ability to manipulate energy in unconventional ways. In this study, we establish a comprehensive theoretical framework to explore exotic wave phenomena in odd elasticity. Specifically, we reveal polarization-dependent energy transfer in the unbroken-phase, second-order amplification at exceptional points (EPs), and exponential energy gain or loss in the broken-phase. To substantiate these findings, we perform an energy-based analysis within a homogeneous medium, offering clear physical interpretations that corroborate our theoretical predictions. Furthermore, we develop a homogenization theory to examine the elastodynamic behavior of various two-dimensional(2D) odd lattices constructed from tensile-torsional asymmetric springs. This theory is validated through comparisons of dispersion relations and coupled mode shapes with those obtained from microstructural simulations. The energy-transfer characteristics and directional wave amplification observed in the lattices closely align with the predictions of the homogenized model. Finally, we demonstrate unidirectional surface wave propagation and mode conversion as striking manifestations of non-Hermitian behavior in mechanical systems. This work provides a broadly applicable framework for investigating non-Hermitian effects in elastic wave systems and introduces novel strategies for the precise manipulation and control of mechanical energy.