Asymptotic strategy for matching homogenized structures. Conductivity problem

The paper is concerned with application of the homogenization theory to bodies containing macroinhomogeneities or bodies, parts of which cannot be homogenized (partial homogenization). This situation arises, in particular, for problems of joining homogeneous and periodically inhomogeneous bodies, or combining inhomogeneous bodies of different periodic structure. The peculiarity of the problem is related to a boundary layer, possibly arising on the interface of the matched components. Moreover, this boundary layer may be either real or fictitious, with the latter occurring due to inaccurate formulation of boundary conditions along the interface, ignoring the effect of the micro-stresses. The consideration is carried out within the framework of the steadystate heat equation. The focus of current investigation is on formulation of the problem for the periodicity cell in case of discontinuous homogenized deformations, when these cannot be treated as independent of the “fast” variables. The first order correctors are constructed. The issue of consistent matching procedure, avoiding emergence of fictitious boundary layers, is discussed. It is shown that the temperature of an inhomogeneous fragment on the boundary may be determined from the solution of the homogenized problem, whereas the derivatives (temperature gradients) require fast correctors of the homogenization theory to be taken into account. The analytical consideration is confirmed by results of numerical simulations.