Application of the asymptotic homogenization in a parametric space to the modeling of structurally heterogeneous materials

Abstract

An asymptotic averaging of differential equations with fast oscillating quasi-periodic coefficients so-called “the asymptotic averaging in parametric space” is developed. The system of equations corresponding to structurally heterogeneous thermoelastic media with smoothly varying microstructures is considered. Such materials are usually treated as functionally graded ones. Their description is satisfied by introducing two types of variables into the coefficients of the system of thermoelastic equations: “fast” and “slow”. The fast variables describe the geometry of the inhomogeneities provided their periodic arrangement. The additional variables allow one to describe a smooth spatial variation of a periodic structure of inclusions, i.e. to describe within the proposed approach various materials with functionally graded properties. In addition, the coefficients may depend on the temperature, which also has a given regular distribution in the space. The allocation of this dependence in the separate parameter makes sense, since it is important in practice. It is shown that the asymptotic averaging solves the smooth aperiodic dependencies for the coefficients of the considered equations system parametrically through the functions of rapid variables. A two-level model system for such structurally heterogeneous materials is formulated as a result of averaging, and an algorithm of accurate definition of thermomechanical properties including the functionally graded and thermally depending ones could be found. The combined numerical-analytical block method based on the Papkovich–Neuber​ representation is proposed for low-level problems with arbitrary structures (e.g. for the problem defined on a cell for functions of fast variables; such a problem determines the effective properties of a material). The structures with spherical and cylindrical inclusions and interphase layers are especially considered and the complete function systems allowing one an efficient solution approximation on a cell with exactly satisfied contact conditions on interlayers is constructed.