Conservative yet constitutively odd elasticity in prestressed metamaterials

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 then inverse-design several uniform 2D solids that act as unique waveguides supporting decoupled modes along special lattice directions: a string-like mode and an exotic, in-plane soft mode with a flexural character ($\omega \sim q^2$). The soft modes are insensitive to the value of the anisotropic prestress and exhibit oscillating momentum density but support a non-oscillatory, constant energy current. This principle of harnessing conservative "oddness" to unlock instability-driven wave phenomena provides a powerful new route to creating tunable materials for filtering, guiding, and controlling mechanical waves.