Elastic properties of anisotropic domain wall lattices

Interest in the elastic properties of regular lattices constructed from domain walls has recently been motivated by cosmological applications as solid dark energy. This work investigates the particularly simple examples of triangular, hexagonal, and square lattices in two dimensions and a variety of more complicated lattices in three dimensions which have cubic symmetry. The relevant rigidity coefficients are computed taking into account nonaffine perturbations where necessary, and these are used to evaluate the propagation velocity for any macroscopic scale perturbation mode. Using this information we assess the stability of the various configurations. It is found that triangular lattices are isotropic and stable, whereas hexagonal lattices are unstable. It is argued that the simple orthonormal cases of a square in two dimensions and the cube in three are stable, except to perturbations of infinite extent. We also find that the more complicated case of a rhombic dodecahedral lattice is stable, except to the existence of transverse modes in certain directions, whereas a lattice formed from truncated octahedra is unstable.