Duality and Sheared Analytic Response in Mechanism-Based Metamaterials

Mechanical metamaterials designed around a zero-energy pathway of deformation known as a mechanism, challenge the conventional picture of elasticity and generate complex spatial response that remains largely uncharted. Here, we present a unified theoretical framework to showing that the presence of a unimode in a 2D structure generates a space of anomalous zero-energy sheared analytic modes. The spatial profiles of these stress-free strain patterns is dual to equilibrium stress configurations. We show a transition at an exceptional point between bulk modes in structures with conventional Poisson ratios (anauxetic) and evanescent surface modes for negative Poisson ratios (auxetic). We suggest a first application of these unusual response properties as a switchable signal amplifier and filter for use in mechanical circuitry and computation.