The homogenized quasi-static model of a thermoelastic composite stitched with reinforcing threads

Abstract

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating many thin inclusions, each of which geometrically is a straight line segment with the endpoints on the body edge. The inclusions (i.e. threads, filaments) are parallel to each other and the problem contains a small positive parameter ϵ, which describes the distance between two neighboring inclusions. Relying on the variational formulation of the problem, we investigate the behavior of solutions as ϵ tends to zero. As the result, we derive a well-posed 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 ϵ→0 is based on the use of the two-scale convergence theory.