High frequency homogenisation for elastic lattices

TL;DR

This paper develops a two-scale asymptotic homogenisation method for elastic lattices, enabling accurate analysis of wave dispersion and effective properties at both low and high frequencies, including complex phenomena like Dirac cones.

Contribution

It introduces a novel homogenisation framework for elastic lattices that captures frequency-dependent wave behavior and complex phenomena, extending beyond scalar lattice models.

Findings

Accurately describes dispersion curves near standing waves.

Demonstrates dynamic anisotropy and Dirac cones in elastic lattices.

Validates the theory with two illustrative 2D lattice examples.

Abstract

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice near the edges of the Brillouin zone. The leading order solution is expressed as a product between the standing wave solution and long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones.