High frequency homogenisation for elastic lattices

AbstractA complete methodology, based on a two-scale asymptotic approach, that enables the ho-mogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distin-guished from scalar lattices in that two or more types of coupled waves exist, even at low frequen-cies. Such a theory enables the determination of e ective material properties at both low and highfrequencies. The theoretical framework is developed for the propagation of waves through latticesof arbitrary geometry and dimension. The asymptotic approach provides a method through whichthe dispersive properties of lattices at frequencies near standing waves can be described; the theoryaccurately describes both the dispersion curves and the response of the lattice near the edges ofthe Brillouin zone. The leading order solution is expressed as a product between the standingwave solution and long-scale envelope functions that are eigensolutions of the homogenised partialdi erential equation. The general theory is supplemented by a pair of illustrative examples fortwo archetypal classes of two-dimensional elastic lattices. The eciency of the asymptotic ap-proach in accurately describing several interesting phenomena is demonstrated, including dynamicanisotropy and Dirac cones.