High-Frequency Asymptotics, Homogenisation and Localisation for Lattices

Abstract

SummaryA two-scales approach, for discrete lattice structures, is developed that uses microscale information to find asymptotic homogenized continuum equations valid on the macroscale. The development recognises the importance of standing waves across an elementary cell of the lattice, on the microscale, and perturbs around the, potentially high frequency, standing wave solutions. For examples of infinite perfect periodic and doubly-periodic lattices the resulting asymptotic equations accurately reproduce the behaviour of all branches of the Bloch spectrum near each of the edges of the Brillouin zone. Lattices in which properties vary slowly upon the macroscale are also considered and the asymptotic technique identifies localised modes that are then compared with numerical simulations. IntroductionDiscrete mass-spring lattice systems form classical models of crystal vibrations in solid state physics (1, 2) and were used by Newton, Kelvin, Born, and many others, to model and interpret wave phenomena and these models were instrumental in the development of wave mechanics; a review of the historical literature is contained in Brillouin's monograph (2). Lattice models remain a valuable and instructive way of modelling and understanding fundamental wave phenomena in crystal lattices and cellular structures (3). A common characteristic behaviour of such models is that they exhibit band-gaps, (4), namely bands of frequencies for which waves do not propagate through the atomic lattice. More recently such discrete models have been used for engineering structures (5) with a view to designing smart structures capable of filtering out various frequencies. These discrete systems, exhibiting band gap behaviour, are closely related to periodic continuous media, for instance photonic crystal fibres (6,7,8), for which band-gaps occur and that have numerous and varied industrial applications; in some limits there is a direct analogy between the discrete and continuum models (9). In the mechanics of cellular structures and lattices, considerable knowledge has been gained about the static and low-frequency behaviour in terms of homogenized models, but comparatively less is known of their high frequency behaviour.A common feature of both discrete and continuum models, that are defect-free and infinite in extent, is that the periodicity of the structure leads to dispersion relations between

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