Odd Elasticity in Strain-Gradient Continua: Anisotropy Classification and Non-Hermitian Elastodynamics

Abstract

Odd elasticity extends continuum mechanics to active nonconservative systems containing internal energy sources, such as robotic active metamaterials with nonreciprocal feedback. Modeling these systems as effective continua requires relaxing Maxwell-Betti reciprocity, which results in stiffness tensors lacking major symmetry. In this contribution, we extend this framework to 3D strain-gradient odd elasticity to incorporate the length scales inherent in these heterogeneous systems. By applying the full hierarchy of 3D point-group symmetries along chiral and achiral branches, we derive the exhaustive set of fourth-, fifth-, and sixth-order odd elastic stiffness tensors. Further, utilizing the dynamic principle of virtual work, we formulate the equilibrium equations, higher-order boundary conditions, and the generalized acoustic tensor for elastodynamic analysis. The results demonstrate that nonreciprocal moduli survive even under full isotropy in the sixth-order stiffness, under hemitropy in the fifth-order stiffness, and under cubic symmetry in fourth-order stiffness. In contrast, the elastodynamic analysis reveals that under isotropic and hemitropic limits, despite the underlying odd stiffness moduli, the acoustic tensor remains unconditionally Hermitian. Non-Hermitian elastodynamics is shown to require anisotropy. By numerically analyzing a transversely hemitropic odd elastic continuum, we demonstrate that this symmetry reduction results in non-Hermitian elastodynamic phenomena including exceptional points in the dispersion spectra and active energy exchange through simultaneous amplification and attenuation of the elastic wave components. These results provide the necessary theoretical foundation for future multiscale homogenization frameworks for anisotropic strain-gradient active microstructures, and for the design of nonreciprocal strain-gradient metamaterials.