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The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter δ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as δ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model. [ABSTRACT FROM AUTHOR]

The paper addresses the mathematical justification of a model describing steady vibrations for a planar thermoelastic body with an incorporated thin inclusion. The body is composed of three parts: two adherents and an adhesive layer between them, and we begin with a general mathematical formulation of a problem. By means of the modern methods of asymptotic analysis, we rigorously investigate the behavior of solutions as the thickness of the adhesive tends to zero. As a result, we construct the model that corresponds to the limit case. It turned out that the adhesive is reduced to the inclusion, which is thin (of zero thickness) and relatively hard (compared to the rigidity of the surrounding body). Furthermore, we supplement the obtained results with numerical experiments demonstrating the consistency of the theoretical conclusions. [ABSTRACT FROM AUTHOR]

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'. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]