Duality and Sheared Analytic Response in Mechanism-Based Metamaterials

Mechanical metamaterials designed around a zero-energy pathway of deformation, known as a mechanism, have repeatedly challenged the conventional picture of elasticity. However, the complex spatial deformations these structures are able to support beyond the uniform mechanism remain largely uncharted. Here we present a unified theoretical framework, showing that the presence of any uniform mechanism in a two-dimensional structure fundamentally changes its elastic response by admitting a family of non-uniform zero-energy deformations. Our formalism reveals a mathematical duality between these stress-free strains, which we term "sheared analytic modes" and the supported spatial profiles of stress. These modes undergo a transition from bulk periodic response to evanescent surface response as the Poisson's ratio $\nu$ of the mechanism is tuned through an exceptional point at $\nu=0$. We suggest a first application of these unusual response properties as a switchable signal amplifier and filter for use in mechanical circuitry and computation.