Exotic Soft Modes in 2D Mechanical Metamaterials Yield Powerful New Analytic Prediction Methods

Maximally Auxetic behavior, where Poisson’s ratio is the most negative, has been explored and identified in 2D perforated elastic sheets in which rigid square elements are connected at the corners by comparatively flexible elastic “hinges”. While these metamaterials are designed to emulate a uniform zero-energy motion of the free hinge material (mechanism), experiments have revealed qualitatively different non-uniform mechanical response. To understand this, we utilize a coarse graining approach, combined with highly detailed finite element simulations and experiments, to reveal that the perforated elastic sheet mechanics is controlled by a novel set of soft modes that correspond precisely to the well-studied planar Conformal Maps. We exploit this very convenient result to demonstrate new and highly accurate methods of analytically solving for linear and non-linear deformations of real materials. This includes a powerful holographic approach, in which large non-linear deformations may be predictably controlled by simple actuation at the boundary. Finally, we introduce a more general methodology for identifying and controlling the soft modes associated with a broad class of 2D mechanisms including the Miura and Eggbox origami patterns.