Abstract
With reference to the homogenization theory, as well as problems of vibrations in physics and mechanics, we note a growing interest in the analysis of domains whose boundaries are singularly perturbed. We consider solids containing large clusters of defects that may possess unusual dynamic responses to external loads. We review the method of the mesoscale uniform asymptotic approximations, which use the Green kernels and take into account different scales (defined relative to the size of individual inclusions), together with boundary layer fields. The method provides an efficient analytical tool, as well as solvers for hybrid numerical schemes aimed at solutions of boundary value and spectral problems for mesoscale structures, containing large clusters of defects. We discuss examples that include solids with many small inclusions, where a small parameter, the relative size of an inclusion, may compete with a large parameter, representing an overall number of inclusions. We note that in some cases, the approach of mesoscale asymptotic approximations provides a powerful alternative to the conventional homogenization algorithms for solids with large clusters of small inclusions.
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