Abstract
The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasi-momentum dependence are found as resolvent limits of the original operator family. This results in asymptotic spectral behaviour with a rich dependence on quasimomenta.
Highlights
The model problem to study time-harmonic waves, with frequency ω, in media with microstructure is −div aε ( x ε )∇u= ω2u in where the wave u represents the information being propagated, such as pressure in acoustics, deformation in elasticity or electromagnetic fields in electromagnetism.1 The microstructured nature of the media is characterised by periodic coefficients aε:2Communicated by F
The convergence of spectra was proved and, by doing so, demonstrates that this β function provides an explicit description of the asymptotic structure of the spectrum. Such an explicit description of the limit spectral behaviour via two-scale homogenisation has made way for mathematical studies of high-contrast media as wave-guides: in [20] using multi-scale asymptotics and supplemented with analysis based on two-scale convergence in [10]
In the context of elasticity, a matrix analogue of the β function is derived and plays the role of frequency-dependent effective density [3,4,32]. Such works demonstrate that the unusual phenomena observed in high-contrast media can be described by non-standard constitutive laws provided via two-scale homogenisation
Summary
The model problem to study time-harmonic waves, with frequency ω, in media with microstructure is. = ω2u in where the wave u represents the information being propagated, such as pressure in acoustics, deformation in elasticity or electromagnetic fields in electromagnetism.. The microstructured nature of the media is characterised by periodic coefficients aε:.
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