Multiscale analysis of stationary thermoelastic vibrations of a composite material

Abstract

The problem of description of stationary vibrations is studied for a planar thermoelastic body incorporating thin inclusions. This problem contains two small positive parameters δ and ε , which describe the thickness of an individual inclusion and the distance between two neighbouring inclusions, respectively. Relying on the variational formulation of the problem, by means of the modern methods of asymptotic analysis, we investigate the behaviour of solutions as δ and ε tend to zero. As the result, we construct two models corresponding to the limit cases. At first, as δ → 0 , by the version of the method of formal asymptotic expansions we derive a limit model in which inclusions are thin (of zero width). Then, from this limit model, as ε → 0 , we derive a homogenized model, which describes effective behaviour on the macroscopic scale, i.e. on the scale where there is no need to take into account each individual inclusion. The limiting passage as ε → 0 is based on the use of the two-scale convergence theory. This article is part of the theme issue ‘Non-smooth variational problems and applications’.