Search

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. The leading order solution is expressed as a product between the standing wave solution and the 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. A significant advantage of the method exposited herein is that it allows one to obtain analytical expressions for the leading order asymptotic solutions and effective material properties, even for complex systems; the methodology allows for greater physical understanding of the behaviour of the system than is usually the case for purely numerical homogenisation schemes. [ABSTRACT FROM AUTHOR]

An asymptotic theory for localised defect states created by altering a single, or finite number, of point masses within a periodic lattice is developed. Asymptotic partial differential equations based upon a high frequency homogenisation methodology are found in one and two dimensions and the frequencies associated with the localised defect states are then compared with exact solutions. In two dimensions, attention is focussed upon a special class of separable lattices that degenerate into coupled one-dimensional systems and upon flat bands in the dispersion diagram. In this latter situation local asymptotic solutions occur which switch from elliptic to hyperbolic behaviour. [ABSTRACT FROM AUTHOR]

In solid-state physics, including photonics and wherever periodic lattice structures occur, it is essential to establish the fundamental features associated with wave propagation through the lattice: This is achieved using Bloch waves, the reciprocal lattice, and the reduction, using periodicity, to consider the irreducible Brillouin zone. A general approach, although widely accepted as not being perfectly legitimate, is to plot the dispersion relations around the edges of the Brillouin zone. We show definitively that this can be dangerous and that an important mode of practical significance is missed if this is done in too cavalier a fashion: This missing mode is illustrated for the design of endoscopes based on spring-mass (discrete) periodic structures and photonic crystals. [ABSTRACT FROM AUTHOR]