Linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance, thermal expansion, stored energy) as well as both first and second statistical moments of stresses in the components are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on Green’s function techniques as well as on the generalization of the “multiparticle effective field” method (MEFM), previously proposed for the estimation of stress field averages in the components. The application of the theory is demonstrated by calculating overall strength surfaces of composite materials. The influence of the coating is analyzed by the use of both the assumption of homogeneity of the stress field in the inclusion core and of the thin-layer hypothesis.
Let us consider statistically uniform random set of coated ellipsoidal inclusions having all the same form, orientation and mechanical properties. We are using the main hypothesis of many micromechanical methods, according to which each inclusion is located inside a homogeneous so-called effective field. It is shown, in the framework of the effective field hypothesis, that from a solution of the pure elastic problem (with zero stress free strains) for the composite the relations for effective thermal expansions, stored energy and average thermoelastic strains inside the components can be found. This way one obtains the generalization of the classical formulae by Rosen and Hashin, which are exact for two-component composites. The proposed theory is applied to the example of composites reinforced with ellipsoidal inclusions with thin inhomogeneous (along inclusion surface) coatings. For a single coated inclusion the micromechanical approach is based on the Green function technique as well as on the interfacial Hill operators.
Functionally graded materials are considered, which consists of a homogeneous matrix and a statistically inhomogeneous random set of ellipsoidal inclusions. The hypothesis of effective field homogeneity near the inclusions is used, non-local effects in the constitutive relations are not considered. Non-local dependencies of local effective elastic properties as well as of conditional averages of the stresses in the components on the local concentration of the inclusions are demonstrated. Numerical results are represented for spherical clusters of spherical inclusions. In the interior of a large cluster, sufficiently far away from the boundary, the local effective moduli coincide with the isotropic effective moduli for the statistically homogeneous medium. However, near the boundary of the cluster the tensors of the effective moduli lose isotropy, i.e., they become transversally isotropic and vary significantly within the boundary layer, the thickness of which equals approximately two diameters of the inclusions (non-local boundary layer effect). The character of the dependence of the effective elastic moduli varies (i.e., they increase or decrease monotonically or non-monotonically with the distance from the boundary of the cluster) with the variation of the cluster size (scale effect). Both average meso stresses and average micro stresses in the phases are estimated along the radius of the cluster.