Homogenization of the static anti-plane shear model for the reinforced composite

Abstract

The static problem of anti-plane shear of a thermoelastic composite, stitched with reinforcing threads, is considered. The original formulation contains a small positive parameter $\varepsilon$, which characterizes the distance between neighboring threads. It is also assumed that the thermomechanical characteristics of the composite body depend on $\varepsilon$. The asymptotic behavior of solutions as the parameter $\varepsilon$ tends to zero is investigated. The limiting transition as $\varepsilon \rightarrow 0+$ is mathematically rigorously justified and represents a homogenization procedure. This transition is based on the application of the standard Allaire--Nguetseng two-scale convergence method and its version by G. Allaire, A. Damlamian, U. Hornung for homogenization on thin inclusions. The result consists of the construction of a limit averaged model of anti-plane shear of the composite material. Using the newly obtained model, numerical experiments are performed, which show consistency of the theoretical conclusions.