Multi-cell homogenization of elastic periodic domains

Recently, much attention has been devoted to the application of homogenization methods for the prediction of the dynamic behavior of periodic domains. One of the most popular techniques consists in the application of the Fourier Transform in space which allows the application of Taylor series approximations for low frequencies/long wavelengths. This method provides continuum equations which approximate the dynamic behavior of the considered periodic domain over a range of frequencies which is defined by the order of the considered Taylor series expansion. This technique is very effective, but suffers from two major drawbacks. First, the order of the Taylor expansion, and therefore the frequency range of approximation, is limited by the corresponding order of the continuum equations and by the number of boundary conditions which may be imposed in accordance with the physical constraints on the system. Second, the approximation at low frequencies does not allow capturing the band gap characteristics of the periodic domain. An attempt at overcoming the latter can be made by applying the Fourier series expansion to a macro cell spanning two (or more) irreducible unit cells of the periodic medium. This multi-cell approach allows the description of both average and intra-cell behavior of the domain, and approximates dispersion relations and corresponding dynamic properties at low frequencies and at frequencies close to the lower band gap. The resulting continuum equations are therefore capable of reproducing in part the band-gap characteristics of the structure. The proposed methodology is tested on simple one-dimensional and two-dimensional structures, which illustrate the method and show its effectiveness.