A three-phase tessellation: solution and effective properties

Two-dimensional, doubly periodic, three-phase structures are considered in the situation where mean fluxes are applied across the structure. The approach is to use complex variables, and to use a mapping that reduces the doubly periodic problem to a much simpler one involving joined sectors. This is a model composite structure in electrostatics (and mathematically analogous areas such as porous media, anti-plane elasticity, heat conduction), and we find various effective parameters and investigate limiting cases. The structure is also amenable to asymptotic methods in the case of highly varying composition and we provide these solutions, partly as a check upon our analysis, and partly as they are useful in their own right

High-frequency homogenization for periodic media

This article is available open access through the publisher’s website at the link below. Copyright @ 2010 The Royal Society.An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.NSERC (Canada) and the EPSRC