Conservative yet constitutively odd elasticity in prestressed metamaterials

Abstract

We introduce a design principle for mechanical metamaterials based on "odd elasticity, once removed." By revisiting classic results relating the variation of Cauchy stress and Lagrangian strain around a prestressed reference state, we show how anisotropic, equilibrium prestress can generate a major anti-symmetry in the material's constitutive response. Tuning the system to such a state drives it to a critical instability, radically transforming its acoustic properties. We demonstrate this by inverse-designing uniform 2D solids that act as unique waveguides supporting decoupled modes along special lattice directions: a string-like mode and an exotic soft mode that is in-plane but has flexural character ($ω\sim q^2$) and exhibits a remarkable "DC" energy current. The magnitude of the anisotropic prestress acts as a control parameter for small-$q$ stability, and, when taken to zero, transforms the string-like mode into another soft mode, giving support for circularly polarized "spin" waves with arbitrary $q$. This principle of harnessing conservative "oddness" to unlock instability-driven wave phenomena provides a powerful new route to creating tunable materials for guiding and controlling mechanical waves.