Conformal elasticity of two-dimensional dilational maximally auxetic materials

The elastic response of a structure depends not only on its material, but on its geometry, as when the thinness of a sheet gives rise to out-of-plane bending fundamentally distinct from the response of a thick slab. Such thinness is similarly exploited in mechanical metamaterials which permit counter-rotations of stiff adjoining elements, ranging from corner-sharing square or triangular pieces to disordered networks. Such behavior can be described via a micromorphic theory which includes these abrupt local rearrangements, giving rise to a long-wavelength elastic theory which resembles conventional Cauchy elasticity with a dramatically reduced bulk modulus. The low-energy deformations consist predominantly of local rotations, translations and dilations without shears and are hence exactly the conformal maps of complex analysis, which admit a simple analytical theory extending even into the nonlinear regime. Despite finite-size effects, bending resistance and disorder, this theory accurately captures response in both finite-element simulations and experimental systems, opening new avenues for shape-changing, programmability and nonlinear response.