The topologically polarized isostatic lattices discovered by Kane and Lubensky (2014, Nat. Phys. 10, 39–45) challenged the standard effective medium theories used in the modeling of many truss-based materials and metamaterials. As a matter of fact, these exhibit Parity (P) asymmetric distributions of zero modes that induce a P-asymmetric elastic behavior, both of which cannot be reproduced within Cauchy elasticity. Here, we propose a new effective medium theory baptized “microtwist elasticity” capable of rendering polarization effects on a macroscopic scale. The theory is valid for trusses on the brink of a polarized-unpolarized phase transition in which case they necessarily exhibit more periodic zero modes than they have dimensions. By mapping each periodic zero mode to a macroscopic degree of freedom, the microtwist theory ends up being a kinematically enriched theory. Microtwist elasticity is constructed thanks to leading order two-scale asymptotics and its constitutive and balance equations are derived for a fairly generic isostatic truss: the Kagome lattice. Various numerical and analytical calculations, of the shape and distribution of zero modes, of dispersion diagrams and of polarization effects, systematically show the quality of the proposed effective medium theory.

凯恩和卢本斯基（2014，Nat。Phys。10，39-45）发现的拓扑极化等静力晶格挑战了用于许多基于桁架的材料和超材料建模的标准有效介质理论。实际上，它们表现出零模式的奇偶性（P）不对称分布，从而引起P不对称的弹性行为，而这两者都无法在柯西弹性范围内再现。在这里，我们提出了一种新的有效的介质理论，即受洗的“微扭转弹性”，能够在宏观尺度上呈现极化效应。该理论对于处于极化-非极化相变边缘的桁架是有效的，在这种情况下，它们必须表现出比尺寸更大的周期性零模。通过将每个周期性零模式映射到宏观自由度，微扭转理论最终成为一种运动学丰富的理论。微扭转弹性是由于前导二阶渐近性而构造的，它的本构和平衡方程是针对相当通用的等静桁架：Kagome晶格推导的。零模的形状和分布，色散图和极化效应的各种数值和分析计算，系统地显示了所提出的有效介质理论的质量。