Homogenisation for hexagonal lattices and honeycomb structures

Abstract

An asymptotic scheme is generated that captures the motion of waves within discrete hexagonal and honeycomb lattices by creating continuum homogenised equations. The accuracy of these continuum effective medium equations in describing the frequency-dependent anisotropy of the lattice structure is demonstrated versus numerical simulations. The general formulation is extended by introducing line defects, often called armchair or zigzag line defects for honeycomb lattices such as graphene, into an otherwise perfect lattice creating surface waves propagating in the direction of the defect and decaying away from it. Further localisation by single defects embedded within the line defect is also considered. Finally the homogenisation of a semi-discrete elastic string structure, consisting of repeated hexagonal cells, is shown to coincide very closely with its discrete lattice counterpart.