A further exploration of the interaction direct derivative (IDD) estimate for the effective properties of multiphase composites taking into account inclusion distribution

The interaction direct derivative (IDD) estimate for the effective properties of composites in the matrix-inclusion type was recently proposed by the authors [1], based on the three-phase model. It has an explicit and almost the simplest structure in comparison with other existing micromechanical estimates, with clear and physically significant explanations to all the involved components. It is universally applicable for various multiphase composites in the matrix-inclusion type, regardless of the material symmetries of matrix, inclusions and effective medium, or the distributions, shapes, orientations, and concentrations of the inclusions. As a sister of our above-mentioned paper, the present one devotes to explore some further fundamental properties of the IDD estimate. It is shown that the IDD estimate is of o(c(2))-accuracy (that is, precise up to the second order of the volume fraction c of the inclusions as c --> 0) whenever all inclusions have ellipsoidal inclusion-matrix cells. With this, we further assess, for the first time, accuracies of other micromechanical estimates. A practical procedure to characterize the spatial distribution of inclusions in the composites is proposed. It is particularly emphasized, based on theoretical analyses and some illustrating examples, that the IDD estimate seems valid for any physically possible high concentration of inclusions; and the concept of the effective stress in the IDD estimate bridges the effective properties and local fields. The latter is required in order to characterize evolutional and irreversible (e.g., damage) mechanical processes.