Accounting for spatial distribution in mean-field homogenization of particulate composites

Abstract

Several analytical mean-field homogenization methods, which take into account the particle volume fraction, shape and orientation are readily available to estimate the effective properties of particulate composites. Models have also been proposed to account for the spatial distribution of the particles. The classical Ponte-Castañeda and Willis (PCW) model is based on a parametrization of the statistical distribution law, while the Interaction Direct Derivative (IDD) model associates a matrix cell to each inclusion, representative of close interactions. In the literature, the use of the IDD is commonly reduced to the particular case of the classical Mori and Tanaka (MT) scheme or to the aforementioned PCW model. The present study introduces an original approach to calibrate the IDD model, for 2D linear conductivity, based on representative images of the microstructure. The links between the models and the range of validity of the IDD model are discussed. Besides, an “IDD-based” PCW model and a two-step scheme are proposed for situations where the IDD estimate is inconsistent (lack of major symmetry). Finally, an image analysis method using Voronoï diagrams is implemented to define the cells associated to each inclusion and supply the models. The method is validated by comparisons between the obtained IDD and PCW estimates, the Mori–Tanaka (MT) model and benchmark full-field numerical simulations. Accounting for the inclusion distribution is seen to lead to better estimates, both qualitatively (by capturing anisotropic behaviors due to the sole distribution) and quantitatively. Possible extensions to elastic composites are discussed.