Multiscale analysis of a model problem of a thermoelastic body with thin inclusions

S. A. Sazhenkov ; I. V. Frankina ; A. I. Furtsev ; P. V. Gilev ; A. G. Gorynin ; O. G. Gorynina ; V. M. Karnaev ; E. I. Leonova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 282-318

Résumé

A model statical problem for a thermoelastic body with thin inclusions is studied. This problem incorporates two small positive parameters $\delta$ and $\varepsilon$, which describe the thickness of an individual inclusion and the distance between two neighboring inclusions, respectively. Relying on the variational formulation of the problem, by means of the modern methods of asymptotic analysis, we investigate the behavior of solutions as $\delta$ and $\varepsilon$ tend to zero. As the result, we construct two models corresponding to the limiting cases. At first, as $\delta \to 0$, we derive a limiting model in which inclusions are thin (of zero diameter). Then, from this limiting model, as $\varepsilon \to 0$, we derive a homogenized model, which describes effective behavior 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 $\varepsilon \to 0$ is based on the use of homogenization theory. The final section of the article presents a series of numerical experiments for the established limiting models.