Solution of general integral equations of micromechanics of heterogeneous materials

One considers a linear composite materials (CM), which consists of a homogeneous matrix containing a random set of heterogeneities. An operator form of solution of the general integral equation (GIE) for the general cases of local and nonlocal problems, static and wave motion phenomena for composite materials with random (statistically homogeneous and inhomogeneous, so-called graded) structures containing coated or uncoated inclusions of any shape and orientation with perfect and imperfect interfaces and subjected to any number of coupled or uncoupled, homogeneous or inhomogeneous external fields of different physical nature. The GIE, connecting the driving fields and fluxes in a point being considered and the fields in the surrounding points, are obtained for the random fields of heterogeneities in the infinite media. Estimations of the effective properties and both the first and second statistical moments of fields in the constituents of CMs are presented in a general form of perturbations introduced by the heterogeneities and taking into account a possible imperfection of interface conditions. The solution methods of GIEs are obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known methods of micromechanics. Some particular cases, asymptotic representations, and simplifications of proposed methods are presented for the particular constitutive equations such as linear thermoelastic cases with the perfect and imperfect interfaces, conductivity problem, problems for piezoelectric and other coupled phenomena, composites with nonlocal elastic properties of constituents, and the wave propagation in composites with electromagnetic, optic and mechanical responses.