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.

机械烧绿石晶格经过实验测试以证明材料的固有极性行为，该材料一侧柔软，另一侧坚硬（Bilal 等人，2017 年）。烧绿石晶格中的拓扑极化要求开发一种新的有效介质理论，因为传统的柯西有效理论无法预测极化现象。在这项研究中，我们开发了烧绿石晶格的 3D 微扭曲有效理论，以捕获 P 不对称零模式，通过该模式，极化在宏观尺度上出现或消失。通过将三个周期性零模式映射到三个宏观自由度，3D 微扭曲理论最终成为一个运动学丰富的理论。3D 微扭曲弹性是通过使用两尺度渐近方法制定的，其本构方程和平衡方程是针对相当通用的等静力晶格推导出来的。所提出的理论的性能通过离散模型的精确解来验证，该模型用于再现零模式和色散关系并定量预测非对称压痕响应。该研究可以阐明传统对称群框架之外的 3D 偏振超材料的新弹性理论，这是以前从未报道过的。