  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. Most notably, the
theory is capable of producing a continuum version of Kane and Lubensky's
topological polarization vector.

Classical Physics;Soft Condensed Matter