Microtwist homogenization of three-dimensional Pyrochlore lattices on zero modes and mechanical polarization

Abstract

The mechanical Pyrochlore lattice was experimentally tested to demonstrate an intrinsically polar behavior of the material, which is soft on one side and hard on the opposite side (Bilal et al., 2017). The topological polarization in Pyrochlore lattices begs for developing a new effective medium theory because conventional Cauchy effective theories cannot predict the polarization phenomenon. In this study, we develop a 3D microtwist effective theory of Pyrochlore lattices to capture the P-asymmetric zero modes by which polarization emerges or fades on a macroscopic scale. By mapping three periodic zero modes to three macroscopic degrees of freedom, the 3D microtwist theory ends up being a kinematically enriched theory. The 3D microtwist elasticity is formulated by using two-scale asymptotic approach and its constitutive and balance equations are derived for a fairly generic isostatic lattice. Performance of the proposed theory is validated by the exact solution of the discrete model for reproducing zero modes and dispersion relations and quantitatively predicting asymmetric indentation responses. The study could shed lights on novel elastic theory of 3D polarized metamaterials outside the conventional framework of symmetry groups, which is never reported before.